• 2.6 Solve Compound Inequalities
  • Introduction
  • 1.1 Use the Language of Algebra
  • 1.2 Integers
  • 1.3 Fractions
  • 1.4 Decimals
  • 1.5 Properties of Real Numbers
  • Key Concepts
  • Review Exercises
  • Practice Test
  • 2.1 Use a General Strategy to Solve Linear Equations
  • 2.2 Use a Problem Solving Strategy
  • 2.3 Solve a Formula for a Specific Variable
  • 2.4 Solve Mixture and Uniform Motion Applications
  • 2.5 Solve Linear Inequalities
  • 2.7 Solve Absolute Value Inequalities
  • 3.1 Graph Linear Equations in Two Variables
  • 3.2 Slope of a Line
  • 3.3 Find the Equation of a Line
  • 3.4 Graph Linear Inequalities in Two Variables
  • 3.5 Relations and Functions
  • 3.6 Graphs of Functions
  • 4.1 Solve Systems of Linear Equations with Two Variables
  • 4.2 Solve Applications with Systems of Equations
  • 4.3 Solve Mixture Applications with Systems of Equations
  • 4.4 Solve Systems of Equations with Three Variables
  • 4.5 Solve Systems of Equations Using Matrices
  • 4.6 Solve Systems of Equations Using Determinants
  • 4.7 Graphing Systems of Linear Inequalities
  • 5.1 Add and Subtract Polynomials
  • 5.2 Properties of Exponents and Scientific Notation
  • 5.3 Multiply Polynomials
  • 5.4 Dividing Polynomials
  • Introduction to Factoring
  • 6.1 Greatest Common Factor and Factor by Grouping
  • 6.2 Factor Trinomials
  • 6.3 Factor Special Products
  • 6.4 General Strategy for Factoring Polynomials
  • 6.5 Polynomial Equations
  • 7.1 Multiply and Divide Rational Expressions
  • 7.2 Add and Subtract Rational Expressions
  • 7.3 Simplify Complex Rational Expressions
  • 7.4 Solve Rational Equations
  • 7.5 Solve Applications with Rational Equations
  • 7.6 Solve Rational Inequalities
  • 8.1 Simplify Expressions with Roots
  • 8.2 Simplify Radical Expressions
  • 8.3 Simplify Rational Exponents
  • 8.4 Add, Subtract, and Multiply Radical Expressions
  • 8.5 Divide Radical Expressions
  • 8.6 Solve Radical Equations
  • 8.7 Use Radicals in Functions
  • 8.8 Use the Complex Number System
  • 9.1 Solve Quadratic Equations Using the Square Root Property
  • 9.2 Solve Quadratic Equations by Completing the Square
  • 9.3 Solve Quadratic Equations Using the Quadratic Formula
  • 9.4 Solve Equations in Quadratic Form
  • 9.5 Solve Applications of Quadratic Equations
  • 9.6 Graph Quadratic Functions Using Properties
  • 9.7 Graph Quadratic Functions Using Transformations
  • 9.8 Solve Quadratic Inequalities
  • 10.1 Finding Composite and Inverse Functions
  • 10.2 Evaluate and Graph Exponential Functions
  • 10.3 Evaluate and Graph Logarithmic Functions
  • 10.4 Use the Properties of Logarithms
  • 10.5 Solve Exponential and Logarithmic Equations
  • 11.1 Distance and Midpoint Formulas; Circles
  • 11.2 Parabolas
  • 11.3 Ellipses
  • 11.4 Hyperbolas
  • 11.5 Solve Systems of Nonlinear Equations
  • 12.1 Sequences
  • 12.2 Arithmetic Sequences
  • 12.3 Geometric Sequences and Series
  • 12.4 Binomial Theorem

Learning Objectives

By the end of this section, you will be able to:

  • Solve compound inequalities with β€œand”
  • Solve compound inequalities with β€œor”
  • Solve applications with compound inequalities

Be Prepared 2.15

Before you get started, take this readiness quiz.

Simplify: 2 5 ( x + 10 ) . 2 5 ( x + 10 ) . If you missed this problem, review Example 1.51 .

Be Prepared 2.16

Simplify: βˆ’ ( x βˆ’ 4 ) . βˆ’ ( x βˆ’ 4 ) . If you missed this problem, review Example 1.54 .

Solve Compound Inequalities with β€œand”

Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word β€œand” or the word β€œor.” For example, the following are compound inequalities.

Compound Inequality

A compound inequality is made up of two inequalities connected by the word β€œand” or the word β€œor.”

To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.

To solve a compound inequality with the word β€œand,” we look for all numbers that make both inequalities true. To solve a compound inequality with the word β€œor,” we look for all numbers that make either inequality true.

Let’s start with the compound inequalities with β€œand.” Our solution will be the numbers that are solutions to both inequalities known as the intersection of the two inequalities. Consider the intersection of two streetsβ€”the part where the streets overlapβ€”belongs to both streets.

To find the solution of the compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphsβ€”where the graphs overlap.

For the compound inequality x > βˆ’3 x > βˆ’3 and x ≀ 2 , x ≀ 2 , we graph each inequality. We then look for where the graphs β€œoverlap”. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See Figure 2.5 .

We can see that the numbers between βˆ’3 βˆ’3 and 2 2 are shaded on both of the first two graphs. They will then be shaded on the solution graph.

The number βˆ’3 βˆ’3 is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.

The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.

This is how we will show our solution in the next examples.

Example 2.61

Solve 6 x βˆ’ 3 < 9 6 x βˆ’ 3 < 9 and 2 x + 9 β‰₯ 3 . 2 x + 9 β‰₯ 3 . Graph the solution and write the solution in interval notation.

Try It 2.121

Solve the compound inequality. Graph the solution and write the solution in interval notation: 4 x βˆ’ 7 < 9 4 x βˆ’ 7 < 9 and 5 x + 8 β‰₯ 3 . 5 x + 8 β‰₯ 3 .

Try It 2.122

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3 x βˆ’ 4 < 5 3 x βˆ’ 4 < 5 and 4 x + 9 β‰₯ 1 . 4 x + 9 β‰₯ 1 .

Solve a compound inequality with β€œand.”

  • Step 1. Solve each inequality.
  • Step 2. Graph each solution. Then graph the numbers that make both inequalities true. This graph shows the solution to the compound inequality.
  • Step 3. Write the solution in interval notation.

Example 2.62

Solve 3 ( 2 x + 5 ) ≀ 18 3 ( 2 x + 5 ) ≀ 18 and 2 ( x βˆ’ 7 ) < βˆ’6 . 2 ( x βˆ’ 7 ) < βˆ’6 . Graph the solution and write the solution in interval notation.

Try It 2.123

Solve the compound inequality. Graph the solution and write the solution in interval notation: 2 ( 3 x + 1 ) ≀ 20 2 ( 3 x + 1 ) ≀ 20 and 4 ( x βˆ’ 1 ) < 2 . 4 ( x βˆ’ 1 ) < 2 .

Try It 2.124

Solve the compound inequality. Graph the solution and write the solution in interval notation: 5 ( 3 x βˆ’ 1 ) ≀ 10 5 ( 3 x βˆ’ 1 ) ≀ 10 and 4 ( x + 3 ) < 8 . 4 ( x + 3 ) < 8 .

Example 2.63

Solve 1 3 x βˆ’ 4 β‰₯ βˆ’2 1 3 x βˆ’ 4 β‰₯ βˆ’2 and βˆ’2 ( x βˆ’ 3 ) β‰₯ 4 . βˆ’2 ( x βˆ’ 3 ) β‰₯ 4 . Graph the solution and write the solution in interval notation.

Try It 2.125

Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 4 x βˆ’ 3 β‰₯ βˆ’1 1 4 x βˆ’ 3 β‰₯ βˆ’1 and βˆ’3 ( x βˆ’ 2 ) β‰₯ 2 . βˆ’3 ( x βˆ’ 2 ) β‰₯ 2 .

Try It 2.126

Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 5 x βˆ’ 5 β‰₯ βˆ’3 1 5 x βˆ’ 5 β‰₯ βˆ’3 and βˆ’4 ( x βˆ’ 1 ) β‰₯ βˆ’2 . βˆ’4 ( x βˆ’ 1 ) β‰₯ βˆ’2 .

Sometimes we have a compound inequality that can be written more concisely. For example, a < x a < x and x < b x < b can be written simply as a < x < b a < x < b and then we call it a double inequality . The two forms are equivalent.

Double Inequality

A double inequality is a compound inequality such as a < x < b . a < x < b . It is equivalent to a < x a < x and x < b . x < b .

To solve a double inequality we perform the same operation on all three β€œparts” of the double inequality with the goal of isolating the variable in the center.

Example 2.64

Solve βˆ’4 ≀ 3 x βˆ’ 7 < 8 . βˆ’4 ≀ 3 x βˆ’ 7 < 8 . Graph the solution and write the solution in interval notation.

When written as a double inequality, 1 ≀ x < 5 , 1 ≀ x < 5 , it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.

Another way to graph the solution of 1 ≀ x < 5 1 ≀ x < 5 is to graph both the solution of x β‰₯ 1 x β‰₯ 1 and the solution of x < 5 . x < 5 . We would then find the numbers that make both inequalities true as we did in previous examples.

Try It 2.127

Solve the compound inequality. Graph the solution and write the solution in interval notation: βˆ’5 ≀ 4 x βˆ’ 1 < 7 . βˆ’5 ≀ 4 x βˆ’ 1 < 7 .

Try It 2.128

Solve the compound inequality. Graph the solution and write the solution in interval notation: βˆ’3 < 2 x βˆ’ 5 ≀ 1 . βˆ’3 < 2 x βˆ’ 5 ≀ 1 .

Solve Compound Inequalities with β€œor”

To solve a compound inequality with β€œor”, we start out just as we did with the compound inequalities with β€œand”—we solve the two inequalities. Then we find all the numbers that make either inequality true.

Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.

To write the solution in interval notation , we will often use the union symbol , βˆͺ βˆͺ to show the union of the solutions shown in the graphs.

Solve a compound inequality with β€œor.”

  • Step 2. Graph each solution. Then graph the numbers that make either inequality true.

Example 2.65

Solve 5 βˆ’ 3 x ≀ βˆ’1 5 βˆ’ 3 x ≀ βˆ’1 or 8 + 2 x ≀ 5 . 8 + 2 x ≀ 5 . Graph the solution and write the solution in interval notation.

Try It 2.129

Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 βˆ’ 2 x ≀ βˆ’3 1 βˆ’ 2 x ≀ βˆ’3 or 7 + 3 x ≀ 4 . 7 + 3 x ≀ 4 .

Try It 2.130

Solve the compound inequality. Graph the solution and write the solution in interval notation: 2 βˆ’ 5 x ≀ βˆ’3 2 βˆ’ 5 x ≀ βˆ’3 or 5 + 2 x ≀ 3 . 5 + 2 x ≀ 3 .

Example 2.66

Solve 2 3 x βˆ’ 4 ≀ 3 2 3 x βˆ’ 4 ≀ 3 or 1 4 ( x + 8 ) β‰₯ βˆ’1 . 1 4 ( x + 8 ) β‰₯ βˆ’1 . Graph the solution and write the solution in interval notation.

Try It 2.131

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3 5 x βˆ’ 7 ≀ βˆ’1 3 5 x βˆ’ 7 ≀ βˆ’1 or 1 3 ( x + 6 ) β‰₯ βˆ’2 . 1 3 ( x + 6 ) β‰₯ βˆ’2 .

Try It 2.132

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3 4 x βˆ’ 3 ≀ 3 3 4 x βˆ’ 3 ≀ 3 or 2 5 ( x + 10 ) β‰₯ 0 . 2 5 ( x + 10 ) β‰₯ 0 .

Solve Applications with Compound Inequalities

Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.

Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a compound inequality . Last, we will solve the compound inequality.

Example 2.67

Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the summer, a property owner will pay $24.72 plus $1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $57.06 and $171.02. How many hcf can the owner use if he wants his usage to stay in the normal range?

Try It 2.133

Due to the drought in California, many communities now have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the summer, a property owner will pay $24.72 plus $1.32 per hcf for Conservation Usage. The bill for Conservation Usage would be between or equal to $31.32 and $52.12. How many hcf can the owner use if she wants her usage to stay in the conservation range?

Try It 2.134

During the winter, a property owner will pay $24.72 plus $1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $49.36 and $86.32. How many hcf will he be allowed to use if he wants his usage to stay in the normal range?

Access this online resource for additional instruction and practice with solving compound inequalities.

  • Compound inequalities

Practice Makes Perfect

In the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.

x < 3 x < 3 and x β‰₯ 1 x β‰₯ 1

x ≀ 4 x ≀ 4 and x > βˆ’2 x > βˆ’2

x β‰₯ βˆ’4 x β‰₯ βˆ’4 and x ≀ βˆ’1 x ≀ βˆ’1

x > βˆ’6 x > βˆ’6 and x < βˆ’3 x < βˆ’3

5 x βˆ’ 2 < 8 5 x βˆ’ 2 < 8 and 6 x + 9 β‰₯ 3 6 x + 9 β‰₯ 3

4 x βˆ’ 1 < 7 4 x βˆ’ 1 < 7 and 2 x + 8 β‰₯ 4 2 x + 8 β‰₯ 4

4 x + 6 ≀ 2 4 x + 6 ≀ 2 and 2 x + 1 β‰₯ βˆ’5 2 x + 1 β‰₯ βˆ’5

4 x βˆ’ 2 ≀ 4 4 x βˆ’ 2 ≀ 4 and 7 x βˆ’ 1 > βˆ’8 7 x βˆ’ 1 > βˆ’8

2 x βˆ’ 11 < 5 2 x βˆ’ 11 < 5 and 3 x βˆ’ 8 > βˆ’5 3 x βˆ’ 8 > βˆ’5

7 x βˆ’ 8 < 6 7 x βˆ’ 8 < 6 and 5 x + 7 > βˆ’3 5 x + 7 > βˆ’3

4 ( 2 x βˆ’ 1 ) ≀ 12 4 ( 2 x βˆ’ 1 ) ≀ 12 and 2 ( x + 1 ) < 4 2 ( x + 1 ) < 4

5 ( 3 x βˆ’ 2 ) ≀ 5 5 ( 3 x βˆ’ 2 ) ≀ 5 and 3 ( x + 3 ) < 3 3 ( x + 3 ) < 3

3 ( 2 x βˆ’ 3 ) > 3 3 ( 2 x βˆ’ 3 ) > 3 and 4 ( x + 5 ) β‰₯ 4 4 ( x + 5 ) β‰₯ 4

βˆ’3 ( x + 4 ) < 0 βˆ’3 ( x + 4 ) < 0 and βˆ’1 ( 3 x βˆ’ 1 ) ≀ 7 βˆ’1 ( 3 x βˆ’ 1 ) ≀ 7

1 2 ( 3 x βˆ’ 4 ) ≀ 1 1 2 ( 3 x βˆ’ 4 ) ≀ 1 and 1 3 ( x + 6 ) ≀ 4 1 3 ( x + 6 ) ≀ 4

3 4 ( x βˆ’ 8 ) ≀ 3 3 4 ( x βˆ’ 8 ) ≀ 3 and 1 5 ( x βˆ’ 5 ) ≀ 3 1 5 ( x βˆ’ 5 ) ≀ 3

5 x βˆ’ 2 ≀ 3 x + 4 5 x βˆ’ 2 ≀ 3 x + 4 and 3 x βˆ’ 4 β‰₯ 2 x + 1 3 x βˆ’ 4 β‰₯ 2 x + 1

3 4 x βˆ’ 5 β‰₯ βˆ’2 3 4 x βˆ’ 5 β‰₯ βˆ’2 and βˆ’3 ( x + 1 ) β‰₯ 6 βˆ’3 ( x + 1 ) β‰₯ 6

2 3 x βˆ’ 6 β‰₯ βˆ’4 2 3 x βˆ’ 6 β‰₯ βˆ’4 and βˆ’4 ( x + 2 ) β‰₯ 0 βˆ’4 ( x + 2 ) β‰₯ 0

1 2 ( x βˆ’ 6 ) + 2 < βˆ’5 1 2 ( x βˆ’ 6 ) + 2 < βˆ’5 and 4 βˆ’ 2 3 x < 6 4 βˆ’ 2 3 x < 6

βˆ’5 ≀ 4 x βˆ’ 1 < 7 βˆ’5 ≀ 4 x βˆ’ 1 < 7

βˆ’3 < 2 x βˆ’ 5 ≀ 1 βˆ’3 < 2 x βˆ’ 5 ≀ 1

5 < 4 x + 1 < 9 5 < 4 x + 1 < 9

βˆ’1 < 3 x + 2 < 8 βˆ’1 < 3 x + 2 < 8

βˆ’8 < 5 x + 2 ≀ βˆ’3 βˆ’8 < 5 x + 2 ≀ βˆ’3

βˆ’6 ≀ 4 x βˆ’ 2 < βˆ’2 βˆ’6 ≀ 4 x βˆ’ 2 < βˆ’2

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

x ≀ βˆ’2 x ≀ βˆ’2 or x > 3 x > 3

x ≀ βˆ’4 x ≀ βˆ’4 or x > βˆ’3 x > βˆ’3

x < 2 x < 2 or x β‰₯ 5 x β‰₯ 5

x < 0 x < 0 or x β‰₯ 4 x β‰₯ 4

2 + 3 x ≀ 4 2 + 3 x ≀ 4 or 5 βˆ’ 2 x ≀ βˆ’1 5 βˆ’ 2 x ≀ βˆ’1

4 βˆ’ 3 x ≀ βˆ’2 4 βˆ’ 3 x ≀ βˆ’2 or 2 x βˆ’ 1 ≀ βˆ’5 2 x βˆ’ 1 ≀ βˆ’5

2 ( 3 x βˆ’ 1 ) < 4 2 ( 3 x βˆ’ 1 ) < 4 or 3 x βˆ’ 5 > 1 3 x βˆ’ 5 > 1

3 ( 2 x βˆ’ 3 ) < βˆ’5 3 ( 2 x βˆ’ 3 ) < βˆ’5 or 4 x βˆ’ 1 > 3 4 x βˆ’ 1 > 3

3 4 x βˆ’ 2 > 4 3 4 x βˆ’ 2 > 4 or 4 ( 2 βˆ’ x ) > 0 4 ( 2 βˆ’ x ) > 0

2 3 x βˆ’ 3 > 5 2 3 x βˆ’ 3 > 5 or 3 ( 5 βˆ’ x ) > 6 3 ( 5 βˆ’ x ) > 6

3 x βˆ’ 2 > 4 3 x βˆ’ 2 > 4 or 5 x βˆ’ 3 ≀ 7 5 x βˆ’ 3 ≀ 7

2 ( x + 3 ) β‰₯ 0 2 ( x + 3 ) β‰₯ 0 or 3 ( x + 4 ) ≀ 6 3 ( x + 4 ) ≀ 6

1 2 x βˆ’ 3 ≀ 4 1 2 x βˆ’ 3 ≀ 4 or 1 3 ( x βˆ’ 6 ) β‰₯ βˆ’2 1 3 ( x βˆ’ 6 ) β‰₯ βˆ’2

3 4 x + 2 ≀ βˆ’1 3 4 x + 2 ≀ βˆ’1 or 1 2 ( x + 8 ) β‰₯ βˆ’3 1 2 ( x + 8 ) β‰₯ βˆ’3

Mixed practice

3 x + 7 ≀ 1 3 x + 7 ≀ 1 and 2 x + 3 β‰₯ βˆ’5 2 x + 3 β‰₯ βˆ’5

6 ( 2 x βˆ’ 1 ) > 6 6 ( 2 x βˆ’ 1 ) > 6 and 5 ( x + 2 ) β‰₯ 0 5 ( x + 2 ) β‰₯ 0

4 βˆ’ 7 x β‰₯ βˆ’3 4 βˆ’ 7 x β‰₯ βˆ’3 or 5 ( x βˆ’ 3 ) + 8 > 3 5 ( x βˆ’ 3 ) + 8 > 3

1 2 x βˆ’ 5 ≀ 3 1 2 x βˆ’ 5 ≀ 3 or 1 4 ( x βˆ’ 8 ) β‰₯ βˆ’3 1 4 ( x βˆ’ 8 ) β‰₯ βˆ’3

βˆ’5 ≀ 2 x βˆ’ 1 < 7 βˆ’5 ≀ 2 x βˆ’ 1 < 7

1 5 ( x βˆ’ 5 ) + 6 < 4 1 5 ( x βˆ’ 5 ) + 6 < 4 and 3 βˆ’ 2 3 x < 5 3 βˆ’ 2 3 x < 5

4 x βˆ’ 2 > 6 4 x βˆ’ 2 > 6 or 3 x βˆ’ 1 ≀ βˆ’2 3 x βˆ’ 1 ≀ βˆ’2

6 x βˆ’ 3 ≀ 1 6 x βˆ’ 3 ≀ 1 and 5 x βˆ’ 1 > βˆ’6 5 x βˆ’ 1 > βˆ’6

βˆ’2 ( 3 x βˆ’ 4 ) ≀ 2 βˆ’2 ( 3 x βˆ’ 4 ) ≀ 2 and βˆ’4 ( x βˆ’ 1 ) < 2 βˆ’4 ( x βˆ’ 1 ) < 2

βˆ’5 ≀ 3 x βˆ’ 2 ≀ 4 βˆ’5 ≀ 3 x βˆ’ 2 ≀ 4

In the following exercises, solve.

Penelope is playing a number game with her sister June. Penelope is thinking of a number and wants June to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Penelope might be thinking of.

Gregory is thinking of a number and he wants his sister Lauren to guess the number. His first clue is that six less than twice his number is between four and forty-two. Write a compound inequality that shows the range of numbers that Gregory might be thinking of.

Andrew is creating a rectangular dog run in his back yard. The length of the dog run is 18 feet. The perimeter of the dog run must be at least 42 feet and no more than 72 feet. Use a compound inequality to find the range of values for the width of the dog run.

Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.

Everyday Math

Blood Pressure A person’s blood pressure is measured with two numbers. The systolic blood pressure measures the pressure of the blood on the arteries as the heart beats. The diastolic blood pressure measures the pressure while the heart is resting.

ⓐ Let x be your systolic blood pressure. Research and then write the compound inequality that shows you what a normal systolic blood pressure should be for someone your age.

β“‘ Let y be your diastolic blood pressure. Research and then write the compound inequality that shows you what a normal diastolic blood pressure should be for someone your age.

Body Mass Index (BMI) is a measure of body fat is determined using your height and weight.

ⓐ Let x be your BMI. Research and then write the compound inequality to show the BMI range for you to be considered normal weight.

β“‘ Research a BMI calculator and determine your BMI. Is it a solution to the inequality in part (a)?

Study Guides > ALGEBRA / TRIG I

Solving compound inequalities, learning outcomes.

  • Solve compound inequalities - OR - express solutions both graphically and with interval notation
  • Solve compound inequalities - AND - express solutions both graphically and with interval notation

Solve Compound Inequalities in the Form of "or"

[latex] \displaystyle \begin{array}{r}x-5>0\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,3x-1<8\,\,\\\underline{\,\,\,+5\,\,+5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+1\,\,+1}\\x\,\,>5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{3x}\,\,\,<\underline{9}\\{3}\,\,\,\,\,\,\,\,\,\,\,\,\,{3}\\x<3\,\,\,\\x>5\,\,\,\,\textit{or}\,\,\,\,x<3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Number line. Open red circle on 3 and red highlight through all numbers less than 3. Open blue circle on 5 and blue highlight on all numbers greater than 5.

[latex] \displaystyle \begin{array}{r}2y+7<13\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,-3y-2\lt 10\\\underline{\,\,\,-7\,\,-7}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+2\,\,\,+2}\\\underline{2y}<\underline{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-3y}<\underline{12}\\{2}\,\,\,\,\,\,\,\,\,\,\,{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-3}\,\,\,\,\,\,\,\,\,\,\,{-3}\\y<3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y\gt -4\\y<3\,\,\,\,\textit{or}\,\,\,\,y\gt -4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Open dot on negative 4 and shaded line going through all numbers greater than negative 4. Open dot on 3 and shaded line on all numbers less than 3. Numbers between closed dot on negative 4 and open dot on 3 are shaded twice.

[latex] \displaystyle \begin{array}{r}5z-3>18\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,-2z-1>15\\\underline{\,\,\,+3\,\,\,+3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+1\,\,\,+1}\\\underline{5z}>\underline{-15}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-2z}>\underline{16}\\{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-2}\,\,\,\,\,\,\,\,\,\,\,{-2}\\z>-3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,z<-8\\z>-3\,\,\,\,\textit{or}\,\,\,\,z<-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Number line. Red open circle on negative 8 and red highlight on all numbers less than negative 8. Open blue circle on negative 3 and blue highlight through all numbers greater than negative 3.

Solve Compound Inequalities in the Form of "and"

[latex] \displaystyle \begin{array}{r}\,\,\,1-4x\le 21\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,5x+2\ge 22\\\underline{-1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,-2\,\,\,\,-2}\\\,\,\,\,\,\underline{-4x}\leq \underline{20}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{5x}\,\,\,\,\,\,\,\ge \underline{20}\\\,\,\,\,\,{-4}\,\,\,\,\,\,\,{-4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\\\,\,\,\,\,\,\,\,\,\,\,x\ge -5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\ge 4\,\,\,\,\\\\x\ge -5\,\text{and}\,\,x\ge 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Number line. Closed blue circle on negative 5 and blue arrow through all numbers greater than negative 5. This blue arrow is labeled x is greater than or equal to negative 5. Closed red circle on 4 and red arrow through all numbers greater than 4. This red line is labeled x is greater than or equal to 4.

[latex] \displaystyle \begin{array}{l}\,\,\,5x-2\le 3\,\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,4x+7>\,\,\,\,3\\\underline{\,\,\,\,\,\,\,\,\,\,\,+2\,\,+2\,}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,\,\,\,\,-7\,\,\,\,\,\,-7}\\\,\,\frac{5x}{5}\,\,\,\,\,\,\,\,\le \frac{5}{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{4x}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,>\frac{-4}{4}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\le 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x>-1\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\le 1\,\,\,\,\text{and}\,\,\,\,x>-1\end{array}[/latex]

solving compound inequalities assignment quizlet

Compound inequalities in the form [latex]a<x<b[/latex]

[latex]\begin{array}{r}\,\,\,\,3\,\,\lt\,\,2x+3\,\,\leq \,\,\,\,7\\\underline{\,-3}\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,-3}\,\,\,\,\,\,\,\,\underline{\,-3}\,\\\,\,\,\,\,\underline{\,0\,}\,\,\lt\,\,\,\,\underline{2x}\,\,\,\,\,\,\,\,\leq\,\,\,\underline{\,4\,}\\2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\\\,\,\,\,\,\,\,\,\,\,0\lt x\leq 2\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Open dot on zero, closed dot on 2, and line through all numbers between zero and two.

[latex] \displaystyle \begin{array}{l}x+2>5\,\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,x+4<5\,\,\,\,\\\underline{\,\,\,\,\,-2\,-2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,-4\,-4}\\x\,\,\,\,\,\,\,\,>\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\,\,\,\,<\,1\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,x>3\,\,\,\,\text{and}\,\,\,\,x<1\end{array}[/latex]

Number line. Red open circle is on 1 and red arrow through all numbers less than 1. Red arrow is labeled x is less than 1. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.

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  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution .

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  • Ex 1: Solve a Compound Inequality Involving AND (Intersection). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution .
  • Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution .

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Compound inequalities are mathematical expressions that involve two or more inequalities combined using the logical operators “AND” or “OR.” These inequalities play a crucial role in solving real-world problems and making informed decisions in various fields such as economics, engineering, and science. In this article, we will explore how to solve compound inequalities step by step, using key terms like “Compound Inequalities,” “Inequality,” and “Solution” to guide us along the way.

What is a Compound Inequality?

A compound inequality is an expression that merges two inequalities, often using “and” or “or.” The two types of compound inequalities are known as conjunction and disjunction, each having its unique characteristics.

Conjunction

Conjunction is a compound inequality where inequalities are joined by “AND.” For example, consider the compound inequality: “-2 < x < 3” or equivalently, “x > -2 AND x < 3.” To solve this type of compound inequality, we need to find values that satisfy both inequalities.

Disjunction

Disjunction is a compound inequality where inequalities are joined by “OR.” For instance, the compound inequality “x ≀ -1 OR x > 2” implies that the solution can contain values that satisfy either or both inequalities. To solve this, we use the “union” symbol (βˆͺ) between the individual solutions.

Step-by-Step Process of Solving Compound Inequalities

Solving compound inequalities involves several steps that are applicable to both conjunction and disjunction.

Step 1: Identify the Inequalities

Start by identifying the two inequalities present in the compound inequality. For example, in the compound inequality “-4 ≀ 3x + 2 < 5,” the two inequalities are “-4 ≀ 3x + 2” and “3x + 2 < 5.”

Step 2: Solve Each Inequality

Solve each inequality separately, just as you would solve a normal inequality. It’s essential to note that when multiplying or dividing an inequality by a negative number, you must reverse the sign of the inequality.

Step 3: Graph the Solutions

Graph the solution of each inequality on the number line. Use open dots to indicate values that are not included and closed dots for values that are included. The direction of the arrow depends on the inequality type.

Step 4: Combine Solutions

Combine the solutions according to whether “AND” or “OR” is specified in the compound inequality.

  • For “AND,” take the intersection of the solutions.
  • For “OR,” take the union of the solutions.

Let’s go through examples to illustrate these steps:

Example of Solving Compound Inequality with “AND”

Consider the compound inequality “-2 < 2x – 3 < 5.”

Method 1: Direct Solution

Start by solving each inequality separately: -2 < 2x – 3″ becomes “1 < 2x. 2x – 3 < 5″ becomes “2x < 8.

Then, divide each inequality by 2: 1/2 < x. x < 4.

Finally, combine the solutions: “1/2 < x < 4” or in interval notation, (1/2, 4).

Method 2: Splitting into Two

Split the compound inequality into two inequalities: 2x – 3 > -2 and 2x – 3 < 5.

Solve each inequality separately: 2x > 1 and 2x < 8.

Divide each inequality by 2: x > 1/2 and x < 4.

Combine the solutions back together: “1/2 < x < 4” or in interval notation, (1/2, 4).

Example of Solving Compound Inequality with “OR”

Consider the compound inequality “2y – 2 ≀ 0 OR 3y β‰₯ 0.”

Solve each inequality separately: “2y – 2 ≀ 0” becomes “2y ≀ 2,” then “y ≀ 1.” “3y β‰₯ 0” implies “y β‰₯ 0.”

Combine the solutions using “OR”: “y ≀ 1 OR y β‰₯ 0.”

The solution is the set of all real numbers, as it covers the entire real number line.

Compound inequalities can sometimes result in “no solution” or represent “the set of all real numbers.” The specific outcome depends on the nature of the inequalities and their logical combination.

In this article, we’ve explored how to solve compound inequalities, which are expressions that involve two or more inequalities combined using “AND” or “OR.” We’ve covered the step-by-step process for solving these inequalities, highlighting key terms such as “Compound Inequalities,” “Inequality,” and “Solution” throughout the discussion. By following these steps, you can effectively tackle compound inequalities in various mathematical and real-world scenarios, enabling you to make informed decisions and solve complex problems.

What is the difference between conjunction and disjunction in compound inequalities?

The main difference lies in the logical connection between the inequalities. In conjunction, represented by “AND,” both inequalities must be satisfied simultaneously for a solution. In disjunction, represented by “OR,” the solution can satisfy either or both inequalities, offering a broader range of possible values.

How do you graph compound inequalities on a number line?

To graph compound inequalities, identify critical points on the number line for each inequality. Use open dots for values not included and closed dots for values included. Draw arrows in the direction specified by the inequality symbols. For “AND,” find the overlapping region; for “OR,” show separate regions.

What is the concept of intersection and union in compound inequalities?

Intersection (∩) combines solutions for “AND,” showing common values that satisfy both inequalities. Union (βˆͺ) combines solutions for “OR,” representing all values that satisfy either or both inequalities, making it more inclusive.

What is the interval notation for compound inequality solutions?

In interval notation, a solution like “1/2 < x < 4” is represented as (1/2, 4). For intervals that include endpoints, use square brackets, e.g., [-7/3, 5/3) means -7/3 ≀ x < 5/3.

Are there any common mistakes to avoid when solving compound inequalities?

Common mistakes include forgetting to reverse the inequality sign when multiplying/dividing by a negative number, incorrectly graphing open and closed dots, and mishandling “AND” and “OR” conditions. Always double-check your work to avoid errors in solving compound inequalities.

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Module 9: Hypothesis Testing With One Sample

Solving compound inequalities, learning outcomes.

By the end of this section, you will be able to:

  • Solve compound inequalities involving β€œor,” and express solutions both graphically and with interval notation
  • Solve compound inequalities involving β€œand,” and express solutions both graphically and with interval notation

Many times we encounter situations with multiple requirements. Sometimes we need only satisfy at least one requirement on a list, and other times we need to satisfy all the requirements on a list. For example, one class may require that you purchase a physical textbook or an e-text. If you have one of these, or both, you have satisfied the requirement. On the other hand, if your class requires you to have a textbook and a calculator, you must have both of these items to satisfy the requirement.

A compound inequality consists of two inequalities joined with the word and or the word or .

Solve Compound Inequalities in the Form of β€œor”

A compound inequality consisting of two inequalities joined with the word or has a solution set made up of all real numbers which satisfy the first inequality, the second inequality, or both. Unions allow us to create a new set from two that may or may not have elements in common.

Some inequalities need to be simplified before their solution can be written or graphed. In the following example, you will see an example of how to solve a one-step inequality in the or form. Note how each inequality is treated independently until the end, where the solution is described in terms of both inequalities. You will use the same properties to solve compound inequalities that you used to solve regular inequalities.

Solve for [latex]x[/latex].

[latex]3x–1<8[/latex] or [latex]x–5>0[/latex]

Solve each inequality by isolating the variable.

[latex] \displaystyle \begin{array}{r}x-5>0\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,3x-1<8\,\,\\\underline{\,\,\,+5\,\,+5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+1\,\,+1}\\x\,\,>5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{3x}\,\,\,<\underline{9}\\{3}\,\,\,\,\,\,\,\,\,\,\,\,\,{3}\\x<3\,\,\,\\x>5\,\,\,\,\textit{or}\,\,\,\,x<3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Inequality notation: [latex] \displaystyle x>5\,\,\,\textit{or}\,\,\,\,x<3[/latex]

Interval notation: [latex]\left(-\infty, 3\right)\cup\left(5,\infty\right)[/latex]

The solution to this compound inequality can also be shown graphically. Sometimes it helps to draw the graph first, before writing the solution, using interval notation.

Number line. Open red circle on 3 and red highlight through all numbers less than 3. Open blue circle on 5 and blue highlight on all numbers greater than 5.

Remember to apply the properties of inequalities when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.

Solve for [latex]y[/latex].

[latex]2y+7\lt13[/latex] or [latex]βˆ’3y–2\lt10[/latex]

Solve each inequality separately.

[latex] \displaystyle \begin{array}{r}2y+7<13\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,-3y-2\lt 10\\\underline{\,\,\,-7\,\,-7}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+2\,\,\,+2}\\\underline{2y}<\underline{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-3y}<\underline{12}\\{2}\,\,\,\,\,\,\,\,\,\,\,{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-3}\,\,\,\,\,\,\,\,\,\,\,{-3}\\y<3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y\gt -4\\y<3\,\,\,\,\textit{or}\,\,\,\,y\gt -4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

The inequality sign is reversed with division by a negative number.

Since y Β could be less thanΒ [latex]3[/latex] or greater than [latex]βˆ’4[/latex], y Β could be any number. Graphing the inequality helps with this interpretation.

Inequality notation: [latex]y<3\text{ or }y> -4[/latex]

Interval notation: [latex]\left(-\infty,\infty\right)[/latex]

Open dot on negative 4 and shaded line going through all numbers greater than negative 4. Open dot on 3 and shaded line on all numbers less than 3. Numbers between closed dot on negative 4 and open dot on 3 are shaded twice.

Even though the graph shows empty dots at [latex]y=3[/latex] and [latex]y=-4[/latex], they are included in the solution.

In the last example, the final answer included solutions whose intervals overlapped. This caused the answer to include all numbers on the number line. In words, we call this solution “all real numbers”.Β Any real number will produce a true statement for eitherΒ [latex]y<3\text{ or }y\gt -4[/latex] when it is substituted for y .

Solve for [latex]z[/latex].

[latex]5z–3\gtβˆ’18[/latex] or [latex]βˆ’2z–1\gt15[/latex]

Solve each inequality separately.Β Combine the solutions.

[latex] \displaystyle \begin{array}{r}5z-3>18\,\,\,\,\,\,\,\,\textit{or}\,\,\,\,\,\,\,\,\,\,-2z-1>15\\\underline{\,\,\,+3\,\,\,+3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,+1\,\,\,+1}\\\underline{5z}>\underline{-15}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-2z}>\underline{16}\\{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{-2}\,\,\,\,\,\,\,\,\,\,\,{-2}\\z>-3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,z<-8\\z>-3\,\,\,\,\textit{or}\,\,\,\,z<-8\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Inequality notation:Β [latex] \displaystyle z>-3\,\,\,\,\textit{or}\,\,\,\,z<-8[/latex]

Interval notation: [latex]\left(-\infty,-8\right)\cup\left(-3,\infty\right)[/latex] Note how we write the intervals with the one containing the most negative solutions first, then move to the right on the number line. [latex]z<-8[/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x>-3[/latex] has solutions that continue all the way to the right.

Number line. Red open circle on negative 8 and red highlight on all numbers less than negative 8. Open blue circle on negative 3 and blue highlight through all numbers greater than negative 3.

The following video contains an example of solving a compound inequality involving orΒ  and drawing the associated graph.

Solve Compound Inequalities in the Form of “and”

The solution of a compound inequality that consists of two inequalities joined with the word and is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an and compound inequality are all the solutions that the two inequalities have in common. As we saw in the last sections, this isΒ where the two graphs overlap.

In this section, we will see more examples where we have to simplify the compound inequalities before we can express their solutions graphically or with an interval.

Solve for x .

[latex] \displaystyle 1-4x\le 21\,\,\,\,\text{and}\,\,\,\,5x+2\ge22[/latex]

Solve each inequality for x .Β Determine the intersection of the solutions.

[latex] \displaystyle \begin{array}{r}\,\,\,1-4x\le 21\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,5x+2\ge 22\\\underline{-1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,-2\,\,\,\,-2}\\\,\,\,\,\,\underline{-4x}\leq \underline{20}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{5x}\,\,\,\,\,\,\,\ge \underline{20}\\\,\,\,\,\,{-4}\,\,\,\,\,\,\,{-4}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{5}\,\,\\\,\,\,\,\,\,\,\,\,\,\,x\ge -5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\ge 4\,\,\,\,\\\\x\ge -5\,\text{and}\,\,x\ge 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

The number line below shows the graphs of the two inequalities in the problem. The solution to the compound inequality is [latex]x\geq4[/latex], sinceΒ this is where the two graphs overlap.

Number line. Closed blue circle on negative 5 and blue arrow through all numbers greater than negative 5. This blue arrow is labeled x is greater than or equal to negative 5. Closed red circle on 4 and red arrow through all numbers greater than 4. This red line is labeled x is greater than or equal to 4.

Inequality: [latex] \displaystyle x\ge 4[/latex]

Interval: [latex]\left[4,\infty\right)[/latex]

Number line. Closed purple circle (overlapping red and blue circles) on 4 and purple arrow through all numbers greater than 4. Purple line is labeled x is greater than or equal to 4.

Solve for x :

[latex] \displaystyle {5}{x}-{2}\le{3}\text{ and }{4}{x}{+7}>{3}[/latex]

Solve each inequality separately.Β Find the overlap between the solutions.

[latex] \displaystyle \begin{array}{l}\,\,\,5x-2\le 3\,\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,4x+7>\,\,\,\,3\\\underline{\,\,\,\,\,\,\,\,\,\,\,+2\,\,+2\,}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,\,\,\,\,-7\,\,\,\,\,\,-7}\\\,\,\frac{5x}{5}\,\,\,\,\,\,\,\,\le \frac{5}{5}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{4x}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,>\frac{-4}{4}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\le 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x>-1\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\le 1\,\,\,\,\text{and}\,\,\,\,x>-1\end{array}[/latex]

Inequality: [latex]-1\le{x}\le{1}[/latex]

Interval: [latex]\left(-1,1\right)[/latex]

x is greater than negative 1 and less than or equal to 1 on a number line. The range is marked on the left with an open dot on negative 1 and on the right with a solid dot on 1.

Compound inequalities in the form [latex]a<x<b[/latex]

Rather than splitting a compound inequality in the form ofΒ Β [latex]a<x<b[/latex]Β into two inequalities [latex]x<b[/latex] and [latex]x>a[/latex], you can more quickly to solve the inequality by applying the properties of inequality to all three segments of the compound inequality.

[latex]3\lt2x+3\leq 7[/latex]

Isolate the variable by subtracting [latex]3[/latex] from all [latex]3[/latex] parts of the inequality, then dividing each part by [latex]2[/latex].

[latex]\begin{array}{r}\,\,\,\,3\,\,\lt\,\,2x+3\,\,\leq \,\,\,\,7\\\underline{\,-3}\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,\,-3}\,\,\,\,\,\,\,\,\underline{\,-3}\,\\\,\,\,\,\,\underline{\,0\,}\,\,\lt\,\,\,\,\underline{2x}\,\,\,\,\,\,\,\,\leq\,\,\,\underline{\,4\,}\\2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\\\,\,\,\,\,\,\,\,\,\,0\lt x\leq 2\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Inequality: [latex] \displaystyle 0\lt{x}\le 2[/latex]

Interval: [latex]\left(0,2\right][/latex]

Open dot on zero, closed dot on 2, and line through all numbers between zero and two.

In the video below, you will see another example of how to solve an inequality in the form [latex]a<x<b[/latex]

To solve inequalities like [latex]a<x<b[/latex], use the addition and multiplication properties of inequality to solve the inequality for x . Whatever operation you perform on the middle portion of the inequality, you must also perform to each of the outside sections as well. Pay particular attention to division or multiplication by a negative.

The solution to a compound inequality with and is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the word and :

In the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.

Solve for x .Β [latex]x+2>5[/latex] and [latex]x+4<5[/latex]

[latex] \displaystyle \begin{array}{l}x+2>5\,\,\,\,\,\,\,\,\,\text{AND}\,\,\,\,\,\,\,x+4<5\,\,\,\,\\\underline{\,\,\,\,\,-2\,-2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,-4\,-4}\\x\,\,\,\,\,\,\,\,>\,\,3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\,\,\,\,<\,1\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,x>3\,\,\,\,\text{and}\,\,\,\,x<1\end{array}[/latex]

Find the overlap between the solutions.

Number line. Red open circle is on 1 and red arrow through all numbers less than 1. Red arrow is labeled x is less than 1. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.

There is no overlap between [latex] \displaystyle x>3[/latex] and [latex]x<1[/latex], so there is no solution.

  • Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
  • Ex 1: Solve a Compound Inequality Involving AND (Intersection). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/UU_KJI59_08 . License : CC BY: Attribution
  • Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. Provided by : Monterey Institute of Technology. Located at : http://nrocnetwork.org/resources/downloads/nroc-math-open-textbook-units-1-12-pdf-and-word-formats/ . License : CC BY: Attribution
  • Question ID 3921, 3920. Authored by : Lippman, D. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL

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Mathematics LibreTexts

2.7: Solve Compound Inequalities

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  • Page ID 114112

Learning Objectives

By the end of this section, you will be able to:

  • Solve compound inequalities with “and”
  • Solve compound inequalities with “or”
  • Solve applications with compound inequalities

Be Prepared 2.15

Before you get started, take this readiness quiz.

Simplify: 2 5 ( x + 10 ) . 2 5 ( x + 10 ) . If you missed this problem, review Example 1.51.

Be Prepared 2.16

Simplify: − ( x − 4 ) . − ( x − 4 ) . If you missed this problem, review Example 1.54.

Solve Compound Inequalities with “and”

Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” For example, the following are compound inequalities.

x + 3 > − 4 and 4 x − 5 ≤ 3 x + 3 > − 4 and 4 x − 5 ≤ 3

2 ( y + 1 ) < 0 or y − 5 ≥ −2 2 ( y + 1 ) < 0 or y − 5 ≥ −2

Compound Inequality

A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”

To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.

To solve a compound inequality with the word “and,” we look for all numbers that make both inequalities true. To solve a compound inequality with the word “or,” we look for all numbers that make either inequality true.

Let’s start with the compound inequalities with “and.” Our solution will be the numbers that are solutions to both inequalities known as the intersection of the two inequalities. Consider the intersection of two streets—the part where the streets overlap—belongs to both streets.

The figure is an illustration of two streets with their intersection shaded

To find the solution of the compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs—where the graphs overlap.

For the compound inequality x > −3 Figure 2.5.

The figure shows the graph of x is greater than negative 3 with a left parenthesis at negative 3 and shading to its right, the graph of x is less than or equal to 2 with a bracket at 2 and shading to its left, and the graph of x is greater than negative 3 and x is less than or equal to 2 with a left parenthesis at negative 3 and a right parenthesis at 2 and shading between negative 3 and 2. Negative 3 and 2 are marked by lines on each number line.

We can see that the numbers between −3 −3 and 2 2 are shaded on both of the first two graphs. They will then be shaded on the solution graph.

The number −3 −3 is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.

The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.

This is how we will show our solution in the next examples.

Example 2.61

Solve 6 x − 3 < 9 6 x − 3 < 9 and 2 x + 9 ≥ 3 . 2 x + 9 ≥ 3 . Graph the solution and write the solution in interval notation.

Try It 2.121

Solve the compound inequality. Graph the solution and write the solution in interval notation: 4 x − 7 < 9 4 x − 7 < 9 and 5 x + 8 ≥ 3 . 5 x + 8 ≥ 3 .

Try It 2.122

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3 x − 4 < 5 3 x − 4 < 5 and 4 x + 9 ≥ 1 . 4 x + 9 ≥ 1 .

Solve a compound inequality with “and.”

  • Step 1. Solve each inequality.
  • Step 2. Graph each solution. Then graph the numbers that make both inequalities true. This graph shows the solution to the compound inequality.
  • Step 3. Write the solution in interval notation.

Example 2.62

Solve 3 ( 2 x + 5 ) ≤ 18 3 ( 2 x + 5 ) ≤ 18 and 2 ( x − 7 ) < −6 . 2 ( x − 7 ) < −6 . Graph the solution and write the solution in interval notation.

Try It 2.123

Solve the compound inequality. Graph the solution and write the solution in interval notation: 2 ( 3 x + 1 ) ≤ 20 2 ( 3 x + 1 ) ≤ 20 and 4 ( x − 1 ) < 2 . 4 ( x − 1 ) < 2 .

Try It 2.124

Solve the compound inequality. Graph the solution and write the solution in interval notation: 5 ( 3 x − 1 ) ≤ 10 5 ( 3 x − 1 ) ≤ 10 and 4 ( x + 3 ) < 8 . 4 ( x + 3 ) < 8 .

Example 2.63

Solve 1 3 x − 4 ≥ −2 1 3 x − 4 ≥ −2 and −2 ( x − 3 ) ≥ 4 . −2 ( x − 3 ) ≥ 4 . Graph the solution and write the solution in interval notation.

Try It 2.125

Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 4 x − 3 ≥ −1 1 4 x − 3 ≥ −1 and −3 ( x − 2 ) ≥ 2 . −3 ( x − 2 ) ≥ 2 .

Try It 2.126

Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 5 x − 5 ≥ −3 1 5 x − 5 ≥ −3 and −4 ( x − 1 ) ≥ −2 . −4 ( x − 1 ) ≥ −2 .

Sometimes we have a compound inequality that can be written more concisely. For example, a < x a < x and x < b x < b can be written simply as a < x < b a < x < b and then we call it a double inequality . The two forms are equivalent.

Double Inequality

A double inequality is a compound inequality such as a < x < b . a < x < b . It is equivalent to a < x a < x and x < b . x < b .

Other forms: a < x < b is equivalent to a < x and x < b a ≤ x ≤ b is equivalent to a ≤ x and x ≤ b a > x > b is equivalent to a > x and x > b a ≥ x ≥ b is equivalent to a ≥ x and x ≥ b Other forms: a < x < b is equivalent to a < x and x < b a ≤ x ≤ b is equivalent to a ≤ x and x ≤ b a > x > b is equivalent to a > x and x > b a ≥ x ≥ b is equivalent to a ≥ x and x ≥ b

To solve a double inequality we perform the same operation on all three “parts” of the double inequality with the goal of isolating the variable in the center.

Example 2.64

Solve −4 ≤ 3 x − 7 < 8 . −4 ≤ 3 x − 7 < 8 . Graph the solution and write the solution in interval notation.

When written as a double inequality, 1 ≤ x < 5 , 1 ≤ x < 5 , it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.

Another way to graph the solution of 1 ≤ x < 5 1 ≤ x < 5 is to graph both the solution of x ≥ 1 x ≥ 1 and the solution of x < 5 . x < 5 . We would then find the numbers that make both inequalities true as we did in previous examples.

Try It 2.127

Solve the compound inequality. Graph the solution and write the solution in interval notation: −5 ≤ 4 x − 1 < 7 . −5 ≤ 4 x − 1 < 7 .

Try It 2.128

Solve the compound inequality. Graph the solution and write the solution in interval notation: −3 < 2 x − 5 ≤ 1 . −3 < 2 x − 5 ≤ 1 .

Solve Compound Inequalities with “or”

To solve a compound inequality with “or”, we start out just as we did with the compound inequalities with “and”—we solve the two inequalities. Then we find all the numbers that make either inequality true.

Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.

To write the solution in interval notation , we will often use the union symbol , ∪ ∪ to show the union of the solutions shown in the graphs.

Solve a compound inequality with “or.”

  • Step 2. Graph each solution. Then graph the numbers that make either inequality true.

Example 2.65

Solve 5 − 3 x ≤ −1 5 − 3 x ≤ −1 or 8 + 2 x ≤ 5 . 8 + 2 x ≤ 5 . Graph the solution and write the solution in interval notation.

Try It 2.129

Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 − 2 x ≤ −3 1 − 2 x ≤ −3 or 7 + 3 x ≤ 4 . 7 + 3 x ≤ 4 .

Try It 2.130

Solve the compound inequality. Graph the solution and write the solution in interval notation: 2 − 5 x ≤ −3 2 − 5 x ≤ −3 or 5 + 2 x ≤ 3 . 5 + 2 x ≤ 3 .

Example 2.66

Solve 2 3 x − 4 ≤ 3 2 3 x − 4 ≤ 3 or 1 4 ( x + 8 ) ≥ −1 . 1 4 ( x + 8 ) ≥ −1 . Graph the solution and write the solution in interval notation.

Try It 2.131

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3 5 x − 7 ≤ −1 3 5 x − 7 ≤ −1 or 1 3 ( x + 6 ) ≥ −2 . 1 3 ( x + 6 ) ≥ −2 .

Try It 2.132

Solve the compound inequality. Graph the solution and write the solution in interval notation: 3 4 x − 3 ≤ 3 3 4 x − 3 ≤ 3 or 2 5 ( x + 10 ) ≥ 0 . 2 5 ( x + 10 ) ≥ 0 .

Solve Applications with Compound Inequalities

Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.

Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a compound inequality . Last, we will solve the compound inequality.

Example 2.67

Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the summer, a property owner will pay $24.72 plus $1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $57.06 and $171.02. How many hcf can the owner use if he wants his usage to stay in the normal range?

Try It 2.133

Due to the drought in California, many communities now have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses.

During the summer, a property owner will pay $24.72 plus $1.32 per hcf for Conservation Usage. The bill for Conservation Usage would be between or equal to $31.32 and $52.12. How many hcf can the owner use if she wants her usage to stay in the conservation range?

Try It 2.134

During the winter, a property owner will pay $24.72 plus $1.54 per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $49.36 and $86.32. How many hcf will he be allowed to use if he wants his usage to stay in the normal range?

Access this online resource for additional instruction and practice with solving compound inequalities.

  • Compound inequalities

Section 2.6 Exercises

Practice makes perfect.

In the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.

x < 3 x < 3 and x ≥ 1 x ≥ 1

x ≤ 4 x ≤ 4 and x > −2 x > −2

x ≥ −4 x ≥ −4 and x ≤ −1 x ≤ −1

x > −6 x > −6 and x < −3 x < −3

5 x − 2 < 8 5 x − 2 < 8 and 6 x + 9 ≥ 3 6 x + 9 ≥ 3

4 x − 1 < 7 4 x − 1 < 7 and 2 x + 8 ≥ 4 2 x + 8 ≥ 4

4 x + 6 ≤ 2 4 x + 6 ≤ 2 and 2 x + 1 ≥ −5 2 x + 1 ≥ −5

4 x − 2 ≤ 4 4 x − 2 ≤ 4 and 7 x − 1 > −8 7 x − 1 > −8

2 x − 11 < 5 2 x − 11 < 5 and 3 x − 8 > −5 3 x − 8 > −5

7 x − 8 < 6 7 x − 8 < 6 and 5 x + 7 > −3 5 x + 7 > −3

4 ( 2 x − 1 ) ≤ 12 4 ( 2 x − 1 ) ≤ 12 and 2 ( x + 1 ) < 4 2 ( x + 1 ) < 4

5 ( 3 x − 2 ) ≤ 5 5 ( 3 x − 2 ) ≤ 5 and 3 ( x + 3 ) < 3 3 ( x + 3 ) < 3

3 ( 2 x − 3 ) > 3 3 ( 2 x − 3 ) > 3 and 4 ( x + 5 ) ≥ 4 4 ( x + 5 ) ≥ 4

−3 ( x + 4 ) < 0 −3 ( x + 4 ) < 0 and −1 ( 3 x − 1 ) ≤ 7 −1 ( 3 x − 1 ) ≤ 7

1 2 ( 3 x − 4 ) ≤ 1 1 2 ( 3 x − 4 ) ≤ 1 and 1 3 ( x + 6 ) ≤ 4 1 3 ( x + 6 ) ≤ 4

3 4 ( x − 8 ) ≤ 3 3 4 ( x − 8 ) ≤ 3 and 1 5 ( x − 5 ) ≤ 3 1 5 ( x − 5 ) ≤ 3

5 x − 2 ≤ 3 x + 4 5 x − 2 ≤ 3 x + 4 and 3 x − 4 ≥ 2 x + 1 3 x − 4 ≥ 2 x + 1

3 4 x − 5 ≥ −2 3 4 x − 5 ≥ −2 and −3 ( x + 1 ) ≥ 6 −3 ( x + 1 ) ≥ 6

2 3 x − 6 ≥ −4 2 3 x − 6 ≥ −4 and −4 ( x + 2 ) ≥ 0 −4 ( x + 2 ) ≥ 0

1 2 ( x − 6 ) + 2 < −5 1 2 ( x − 6 ) + 2 < −5 and 4 − 2 3 x < 6 4 − 2 3 x < 6

−5 ≤ 4 x − 1 < 7 −5 ≤ 4 x − 1 < 7

−3 < 2 x − 5 ≤ 1 −3 < 2 x − 5 ≤ 1

5 < 4 x + 1 < 9 5 < 4 x + 1 < 9

−1 < 3 x + 2 < 8 −1 < 3 x + 2 < 8

−8 < 5 x + 2 ≤ −3 −8 < 5 x + 2 ≤ −3

−6 ≤ 4 x − 2 < −2 −6 ≤ 4 x − 2 < −2

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

x ≤ −2 x ≤ −2 or x > 3 x > 3

x ≤ −4 x ≤ −4 or x > −3 x > −3

x < 2 x < 2 or x ≥ 5 x ≥ 5

x < 0 x < 0 or x ≥ 4 x ≥ 4

2 + 3 x ≤ 4 2 + 3 x ≤ 4 or 5 − 2 x ≤ −1 5 − 2 x ≤ −1

4 − 3 x ≤ −2 4 − 3 x ≤ −2 or 2 x − 1 ≤ −5 2 x − 1 ≤ −5

2 ( 3 x − 1 ) < 4 2 ( 3 x − 1 ) < 4 or 3 x − 5 > 1 3 x − 5 > 1

3 ( 2 x − 3 ) < −5 3 ( 2 x − 3 ) < −5 or 4 x − 1 > 3 4 x − 1 > 3

3 4 x − 2 > 4 3 4 x − 2 > 4 or 4 ( 2 − x ) > 0 4 ( 2 − x ) > 0

2 3 x − 3 > 5 2 3 x − 3 > 5 or 3 ( 5 − x ) > 6 3 ( 5 − x ) > 6

3 x − 2 > 4 3 x − 2 > 4 or 5 x − 3 ≤ 7 5 x − 3 ≤ 7

2 ( x + 3 ) ≥ 0 2 ( x + 3 ) ≥ 0 or 3 ( x + 4 ) ≤ 6 3 ( x + 4 ) ≤ 6

1 2 x − 3 ≤ 4 1 2 x − 3 ≤ 4 or 1 3 ( x − 6 ) ≥ −2 1 3 ( x − 6 ) ≥ −2

3 4 x + 2 ≤ −1 3 4 x + 2 ≤ −1 or 1 2 ( x + 8 ) ≥ −3 1 2 ( x + 8 ) ≥ −3

Mixed practice

3 x + 7 ≤ 1 3 x + 7 ≤ 1 and 2 x + 3 ≥ −5 2 x + 3 ≥ −5

6 ( 2 x − 1 ) > 6 6 ( 2 x − 1 ) > 6 and 5 ( x + 2 ) ≥ 0 5 ( x + 2 ) ≥ 0

4 − 7 x ≥ −3 4 − 7 x ≥ −3 or 5 ( x − 3 ) + 8 > 3 5 ( x − 3 ) + 8 > 3

1 2 x − 5 ≤ 3 1 2 x − 5 ≤ 3 or 1 4 ( x − 8 ) ≥ −3 1 4 ( x − 8 ) ≥ −3

−5 ≤ 2 x − 1 < 7 −5 ≤ 2 x − 1 < 7

1 5 ( x − 5 ) + 6 < 4 1 5 ( x − 5 ) + 6 < 4 and 3 − 2 3 x < 5 3 − 2 3 x < 5

4 x − 2 > 6 4 x − 2 > 6 or 3 x − 1 ≤ −2 3 x − 1 ≤ −2

6 x − 3 ≤ 1 6 x − 3 ≤ 1 and 5 x − 1 > −6 5 x − 1 > −6

−2 ( 3 x − 4 ) ≤ 2 −2 ( 3 x − 4 ) ≤ 2 and −4 ( x − 1 ) < 2 −4 ( x − 1 ) < 2

−5 ≤ 3 x − 2 ≤ 4 −5 ≤ 3 x − 2 ≤ 4

In the following exercises, solve.

Penelope is playing a number game with her sister June. Penelope is thinking of a number and wants June to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Penelope might be thinking of.

Gregory is thinking of a number and he wants his sister Lauren to guess the number. His first clue is that six less than twice his number is between four and forty-two. Write a compound inequality that shows the range of numbers that Gregory might be thinking of.

Andrew is creating a rectangular dog run in his back yard. The length of the dog run is 18 feet. The perimeter of the dog run must be at least 42 feet and no more than 72 feet. Use a compound inequality to find the range of values for the width of the dog run.

Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.

Everyday Math

Blood Pressure A person’s blood pressure is measured with two numbers. The systolic blood pressure measures the pressure of the blood on the arteries as the heart beats. The diastolic blood pressure measures the pressure while the heart is resting.

ⓐ Let x be your systolic blood pressure. Research and then write the compound inequality that shows you what a normal systolic blood pressure should be for someone your age.

ⓑ Let y be your diastolic blood pressure. Research and then write the compound inequality that shows you what a normal diastolic blood pressure should be for someone your age.

Body Mass Index (BMI) is a measure of body fat is determined using your height and weight.

ⓐ Let x be your BMI. Research and then write the compound inequality to show the BMI range for you to be considered normal weight.

ⓑ Research a BMI calculator and determine your BMI. Is it a solution to the inequality in part (a)?

Writing Exercises

In your own words, explain the difference between the properties of equality and the properties of inequality.

Explain the steps for solving the compound inequality 2 − 7 x ≥ −5 2 − 7 x ≥ −5 or 4 ( x − 3 ) + 7 > 3 . 4 ( x − 3 ) + 7 > 3 .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and four rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was solve compound inequalities with “and.” In row 3, the I can was solve compound inequalities with “or.” In row 4, the I can was solve applications with compound inequalities.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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Solving Compound Inequalities

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  • 1. Multiple Choice 3 minutes 1 pt x+2<-4 or -5x<-15 -6<x<3 x<-3 and x>6 x<-6 or x>3 No Solution
  • 3. Multiple Choice 3 minutes 1 pt 0 < x + 7 < 9 -7<x   and   x<2 -7<x   or   x<2 -7>x   and   x>2 -7>x  or   x>2
  • 7. Multiple Choice 3 minutes 1 pt -8 < 2(x + 4) or -3x + 4 > x - 4 No Solution x < -8 or x < 2 x > -8 or x < 2 x < -8 and x < 2
  • 8. Multiple Choice 3 minutes 1 pt 3x-4 < -13 or 7x+1  > 22 All Real numbers   x > 3   and x > -9 The answer is not listed x < -3   or x > 3
  • 9. Multiple Choice 2 minutes 1 pt Which number is not a possible solution to the compound inequality -5 < x β‰€ 0 -3 -1 0 -5
  • 10. Multiple Choice 2 minutes 1 pt Which number is not a possible solution for the compound inequality:  x > 8 or x < -1 10 5 -5 -10
  • 11. Multiple Choice 3 minutes 1 pt Solve the compound inequality: -2 < 3x -5 β‰€ 4 -1 < x ≀ 3 1 β‰€ x β‰€ 7 1 < x β‰€ 3 -2/3 < x ≀ 4/3
  • 12. Multiple Choice 3 minutes 1 pt Solve the compound inequality:   2x - 4 > 8  or  3x - 1 < -10 x < 6 or x < -3 -3 < x < 6 x > 2 or x < -3 x > 6 or x < -3
  • 13. Multiple Choice 3 minutes 1 pt Solve: 6x - 14 < 14 or 3x + 10 > 13  x < -4 or x > 7 x < -6 or x > 0 x < 0 or x > 1 x < 0 or x > 7
  • 14. Multiple Choice 3 minutes 1 pt Solve:  2x - 8 ≀ 4  AND x + 5 β‰₯ 7 2 < x < 6 x β‰€ 2 or x β‰₯ 6 x < 2 or x > 6 2 β‰€ x β‰€ 6

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IMAGES

  1. How to Solve Compound Inequalities in 3 Easy Steps

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  2. Solving Systems of Inequalities Set Flashcards

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  3. 2.5 Creating and Solving Compound Inequalities Flashcards

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  4. Solving Compound Inequalities Worksheet

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  5. Compound Inequalities Worksheets with Answer Key

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  6. Compound Inequalities, Intersections

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VIDEO

  1. 1 7 Compound Inequalities & Graphing Linear Inequalities with Two Variables

  2. 4.2 Solving Compound Inequalities Involving AND or OR

  3. Solving Compound Inequalities

  4. Graphing Linear Inequalities Assignment (reupload)

  5. Compound Inequalities

  6. Compound Inequalities

COMMENTS

  1. SOLVING COMPOUND INEQUALITIES Flashcards

    SOLVING COMPOUND INEQUALITIES 4.6 (34 reviews) compound inequality Click the card to flip πŸ‘† a statement formed by two or more inequalities Click the card to flip πŸ‘† 1 / 9 Flashcards Learn Test Match Q-Chat Created by smartguy12345678910 Students also viewed Math unit 4 quiz 2 15 terms E-A-vdw Preview Algebra (ABSOLUTE VALUE SOLUTION SETS) 12 terms

  2. Solving Compound Inequalities Flashcards

    Solving Compound Inequalities Flashcards | Quizlet Solving Compound Inequalities -10 < x < 8 Click the card to flip πŸ‘† Solve the Inequality. -1 < 9 + x < 17 Click the card to flip πŸ‘† 1 / 43 Flashcards Learn Test Match Q-Chat Created by Gail_Mosier Teacher Students also viewed River Valley Civilizations Teacher 24 terms Lori_Montone Preview

  3. Introduction to Compound Inequalities: Assignment

    The solution set of the "or" compound inequality contains values for x that satisfy either or both inequalities, which includes all real numbers. Study with Quizlet and memorize flashcards containing terms like Which set of numbers is included as part of the solution set of the compound inequalityx < 6 or x > 10?, The compound inequality 8.00 ...

  4. Quiz 2: Compound Inequalities Flashcards

    1 / 17 Flashcards Learn Test Match Q-Chat jlclark198 Top creator on Quizlet Includes lessons Compound Inequality Graphs, Solving Compound Inequalities, and Inequalities with Two Variables. Students also viewed Math unit 4 quiz 2 15 terms E-A-vdw Preview Algebra I section 4. INEQUALITIES QUIZ COMPOUND INEQUALITIES 15 terms DatSmartBoi_123321 Preview

  5. Inequalities / Assignment Flashcards

    1 / 11 Flashcards Learn Test Match Q-Chat Created by kennedywhite2812 Teacher Students also viewed Inequalities 16 terms Korina_Madrid8 Preview inequalities 18 terms nathanw287 Preview relations & functions assignment 17 terms ashtynkassidee Preview Functional Groups 7 terms need2studyyyy Preview function operations assignment 12 terms

  6. Solving Compound Inequalities Flashcards

    Study with Quizlet and memorize flashcards containing terms like 3<x-1<7, 4<2x+2<8, 10<-2x<22 and more.

  7. Solving Compound Inequalities! Flashcards

    Solving Compound Inequalities! 5 < x < 7 Click the card to flip πŸ‘† 8 < x + 3 < 10 Click the card to flip πŸ‘† 1 / 12 Flashcards Learn Test Match Q-Chat Created by Amity_Kelsch Teacher Terms in this set (12) 5 < x < 7 8 < x + 3 < 10 -5 < x < 7 -6 < x - 1 < 6 x < 4 or x > 8 6x - 15 > 9 or 10x > 80 10 < x < 12 37 < 3x + 7 < 43 x < - 2 or x > 3

  8. 2.6 Solve Compound Inequalities

    To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.

  9. Solve Compound Inequalities

    In this section we will learn how to solve compound inequalities that are joined with the words AND and OR. First, it will help to see some examples of inequalities, intervals, and graphs of compound inequalities. This will help you describe the solutions to compound inequalities properly.

  10. Compound inequalities review (article)

    Solving compound inequalities Example with "OR" Solve for x . 2 x + 3 β‰₯ 7 OR 2 x + 9 > 11 Solving the first inequality for x , we get: 2 x + 3 β‰₯ 7 2 x β‰₯ 4 x β‰₯ 2 Solving the second inequality for x , we get: 2 x + 9 > 11 2 x > 2 x > 1 Graphically, we get: So our compound inequality can be expressed as the simple inequality: x > 1

  11. Compound inequalities

    You might need: Calculator Solve for x . βˆ’ 7 x βˆ’ 50 ≀ βˆ’ 1 AND βˆ’ 6 x + 70 > βˆ’ 2 Choose 1 answer: x β‰₯ βˆ’ 7 A x β‰₯ βˆ’ 7 βˆ’ 7 ≀ x < 12 B βˆ’ 7 ≀ x < 12 x < 12 C x < 12 There are no solutions D There are no solutions All values of x are solutions E All values of x are solutions Show Calculator Stuck? Review related articles/videos or use a hint.

  12. 2.7: Solve Compound Inequalities

    Write the solution in interval notation. [ βˆ’ 3, 2) All the numbers that make both inequalities true are the solution to the compound inequality. Try It 2.7. 2. Solve the compound inequality. Graph the solution and write the solution in interval notation: 4 x βˆ’ 7 < 9 and 5 x + 8 β‰₯ 3. Answer.

  13. Compound inequalities: AND

    To solve a compound inequality, you start by solving each individual inequality. Then, the word "AND" or "OR" tells you the next step to take. AND tells you to find the intersection of the two solution sets. An intersection is the values in common or the overlap of the two sets. This is why it is common to graph the 2 original inequalities.

  14. 6.2.3 Solving Compound Inequalities

    The solution of a compound inequality that consists of two inequalities joined with the word or is the union of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common. The following example shows how to solve a one-step inequality in the or form.

  15. Solving Compound Inequalities

    How to Solve Compound Inequalities When solving compound inequalities, we are going to deal with two general cases or types. The first case involves solving two linear inequalities joined by the word "and". The word "and" is also known as a conjunction.

  16. 1.6: Solve Compound Inequalities

    The number line below shows the graphs of the two inequalities in the problem. The solution to the compound inequality is x β‰₯ 4, since this is where the two graphs overlap. Answer. Inequality: x β‰₯ 4. Interval: [ 4, ∞) Graph: [/hidden-answer] Example. Solve for x: 5 x βˆ’ 2 ≀ 3 and 4 x + 7 > 3.

  17. Study Guide

    Solve Compound Inequalities in the Form of "and". The solution of a compound inequality that consists of two inequalities joined with the word and is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an and compound inequality are all the solutions that the two ...

  18. How to Solve Compound Inequalities? Step-by-Step Explanation

    What is a Compound Inequality? Start by solving each inequality separately: -2 < 2x - 3β€³ becomes "1 < 2x. 2x - 3 < 5β€³ becomes "2x < 8. Then, divide each inequality by 2: Solve each inequality separately: Solve each inequality separately: "2y - 2 ≀ 0" becomes "2y ≀ 2," then "y ≀ 1." "3y β‰₯ 0" implies "y β‰₯ 0." About us Contact us

  19. Solving Compound Inequalities

    Solve Compound Inequalities in the Form of "or". A compound inequality consisting of two inequalities joined with the word or has a solution set made up of all real numbers which satisfy the first inequality, the second inequality, or both. Unions allow us to create a new set from two that may or may not have elements in common.

  20. 1.6: Solve Compound Inequalities

    Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 5x βˆ’ 5 β‰₯ βˆ’ 3 and βˆ’ 4(x βˆ’ 1) β‰₯ βˆ’ 2. Answer. Sometimes we have a compound inequality that can be written more concisely. For example, a < x and x < b can be written simply as a < x < b and then we call it a double inequality.

  21. Solving inequalities: Quiz 1

    Quiz 1. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

  22. 2.7: Solve Compound Inequalities

    Solve Compound Inequalities with "or" To solve a compound inequality with "or", we start out just as we did with the compound inequalities with "and"β€”we solve the two inequalities. Then we find all the numbers that make either inequality true.. Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either ...

  23. Solving Compound Inequalities

    Solving Compound Inequalities Julie Bensley 1.3K plays 14 questions Copy & Edit Live Session Assign Show Answers See Preview 1. Multiple Choice 3 minutes 1 pt x+2<-4 or -5x<-15 -6<x<3 x<-3 and x>6 x<-6 or x>3 No Solution 2. Multiple Choice 3 minutes 1 pt Solve: A B C D 3. Multiple Choice 3 minutes 1 pt 0 < x + 7 < 9 -7<x and x<2 -7<x or x<2