- 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 βandβ
- Solve compound inequalities with βorβ
- 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: β ( x β 4 ) . β ( x β 4 ) . If you missed this problem, review Example 1.54 .
Solve Compound Inequalities with βandβ
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 βandβ or the word βor.β For example, the following are compound inequalities.
Compound Inequality
A compound inequality is made up of two inequalities connected by the word βandβ or the word βor.β
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 βand,β we look for all numbers that make both inequalities true. To solve a compound inequality with the word βor,β we look for all numbers that make either inequality true.
Letβs start with the compound inequalities with βand.β 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 streetsβthe part where the streets overlapβbelongs 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 graphsβwhere the graphs overlap.
For the compound inequality x > β3 x > β3 and x β€ 2 , x β€ 2 , we graph each inequality. We then look for where the graphs βoverlapβ. 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 β3 β3 and 2 2 are shaded on both of the first two graphs. They will then be shaded on the solution graph.
The number β3 β3 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 β 3 < 9 6 x β 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 β 7 < 9 4 x β 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 β 4 < 5 3 x β 4 < 5 and 4 x + 9 β₯ 1 . 4 x + 9 β₯ 1 .
Solve a compound inequality with βand.β
- 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 ) β€ 18 3 ( 2 x + 5 ) β€ 18 and 2 ( x β 7 ) < β6 . 2 ( x β 7 ) < β6 . 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 ) β€ 20 2 ( 3 x + 1 ) β€ 20 and 4 ( x β 1 ) < 2 . 4 ( x β 1 ) < 2 .
Try It 2.124
Solve the compound inequality. Graph the solution and write the solution in interval notation: 5 ( 3 x β 1 ) β€ 10 5 ( 3 x β 1 ) β€ 10 and 4 ( x + 3 ) < 8 . 4 ( x + 3 ) < 8 .
Example 2.63
Solve 1 3 x β 4 β₯ β2 1 3 x β 4 β₯ β2 and β2 ( x β 3 ) β₯ 4 . β2 ( x β 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 β 3 β₯ β1 1 4 x β 3 β₯ β1 and β3 ( x β 2 ) β₯ 2 . β3 ( x β 2 ) β₯ 2 .
Try It 2.126
Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 5 x β 5 β₯ β3 1 5 x β 5 β₯ β3 and β4 ( x β 1 ) β₯ β2 . β4 ( x β 1 ) β₯ β2 .
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 βpartsβ of the double inequality with the goal of isolating the variable in the center.
Example 2.64
Solve β4 β€ 3 x β 7 < 8 . β4 β€ 3 x β 7 < 8 . Graph the solution and write the solution in interval notation.
When written as a double inequality, 1 β€ x < 5 , 1 β€ 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 β€ x < 5 1 β€ 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: β5 β€ 4 x β 1 < 7 . β5 β€ 4 x β 1 < 7 .
Try It 2.128
Solve the compound inequality. Graph the solution and write the solution in interval notation: β3 < 2 x β 5 β€ 1 . β3 < 2 x β 5 β€ 1 .
Solve Compound Inequalities with βorβ
To solve a compound inequality with βorβ, we start out just as we did with the compound inequalities with βandββwe 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 , βͺ βͺ to show the union of the solutions shown in the graphs.
Solve a compound inequality with βor.β
- Step 2. Graph each solution. Then graph the numbers that make either inequality true.
Example 2.65
Solve 5 β 3 x β€ β1 5 β 3 x β€ β1 or 8 + 2 x β€ 5 . 8 + 2 x β€ 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 β 2 x β€ β3 1 β 2 x β€ β3 or 7 + 3 x β€ 4 . 7 + 3 x β€ 4 .
Try It 2.130
Solve the compound inequality. Graph the solution and write the solution in interval notation: 2 β 5 x β€ β3 2 β 5 x β€ β3 or 5 + 2 x β€ 3 . 5 + 2 x β€ 3 .
Example 2.66
Solve 2 3 x β 4 β€ 3 2 3 x β 4 β€ 3 or 1 4 ( x + 8 ) β₯ β1 . 1 4 ( x + 8 ) β₯ β1 . 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 β 7 β€ β1 3 5 x β 7 β€ β1 or 1 3 ( x + 6 ) β₯ β2 . 1 3 ( x + 6 ) β₯ β2 .
Try It 2.132
Solve the compound inequality. Graph the solution and write the solution in interval notation: 3 4 x β 3 β€ 3 3 4 x β 3 β€ 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 β€ 4 x β€ 4 and x > β2 x > β2
x β₯ β4 x β₯ β4 and x β€ β1 x β€ β1
x > β6 x > β6 and x < β3 x < β3
5 x β 2 < 8 5 x β 2 < 8 and 6 x + 9 β₯ 3 6 x + 9 β₯ 3
4 x β 1 < 7 4 x β 1 < 7 and 2 x + 8 β₯ 4 2 x + 8 β₯ 4
4 x + 6 β€ 2 4 x + 6 β€ 2 and 2 x + 1 β₯ β5 2 x + 1 β₯ β5
4 x β 2 β€ 4 4 x β 2 β€ 4 and 7 x β 1 > β8 7 x β 1 > β8
2 x β 11 < 5 2 x β 11 < 5 and 3 x β 8 > β5 3 x β 8 > β5
7 x β 8 < 6 7 x β 8 < 6 and 5 x + 7 > β3 5 x + 7 > β3
4 ( 2 x β 1 ) β€ 12 4 ( 2 x β 1 ) β€ 12 and 2 ( x + 1 ) < 4 2 ( x + 1 ) < 4
5 ( 3 x β 2 ) β€ 5 5 ( 3 x β 2 ) β€ 5 and 3 ( x + 3 ) < 3 3 ( x + 3 ) < 3
3 ( 2 x β 3 ) > 3 3 ( 2 x β 3 ) > 3 and 4 ( x + 5 ) β₯ 4 4 ( x + 5 ) β₯ 4
β3 ( x + 4 ) < 0 β3 ( x + 4 ) < 0 and β1 ( 3 x β 1 ) β€ 7 β1 ( 3 x β 1 ) β€ 7
1 2 ( 3 x β 4 ) β€ 1 1 2 ( 3 x β 4 ) β€ 1 and 1 3 ( x + 6 ) β€ 4 1 3 ( x + 6 ) β€ 4
3 4 ( x β 8 ) β€ 3 3 4 ( x β 8 ) β€ 3 and 1 5 ( x β 5 ) β€ 3 1 5 ( x β 5 ) β€ 3
5 x β 2 β€ 3 x + 4 5 x β 2 β€ 3 x + 4 and 3 x β 4 β₯ 2 x + 1 3 x β 4 β₯ 2 x + 1
3 4 x β 5 β₯ β2 3 4 x β 5 β₯ β2 and β3 ( x + 1 ) β₯ 6 β3 ( x + 1 ) β₯ 6
2 3 x β 6 β₯ β4 2 3 x β 6 β₯ β4 and β4 ( x + 2 ) β₯ 0 β4 ( x + 2 ) β₯ 0
1 2 ( x β 6 ) + 2 < β5 1 2 ( x β 6 ) + 2 < β5 and 4 β 2 3 x < 6 4 β 2 3 x < 6
β5 β€ 4 x β 1 < 7 β5 β€ 4 x β 1 < 7
β3 < 2 x β 5 β€ 1 β3 < 2 x β 5 β€ 1
5 < 4 x + 1 < 9 5 < 4 x + 1 < 9
β1 < 3 x + 2 < 8 β1 < 3 x + 2 < 8
β8 < 5 x + 2 β€ β3 β8 < 5 x + 2 β€ β3
β6 β€ 4 x β 2 < β2 β6 β€ 4 x β 2 < β2
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
x β€ β2 x β€ β2 or x > 3 x > 3
x β€ β4 x β€ β4 or x > β3 x > β3
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 β€ 4 2 + 3 x β€ 4 or 5 β 2 x β€ β1 5 β 2 x β€ β1
4 β 3 x β€ β2 4 β 3 x β€ β2 or 2 x β 1 β€ β5 2 x β 1 β€ β5
2 ( 3 x β 1 ) < 4 2 ( 3 x β 1 ) < 4 or 3 x β 5 > 1 3 x β 5 > 1
3 ( 2 x β 3 ) < β5 3 ( 2 x β 3 ) < β5 or 4 x β 1 > 3 4 x β 1 > 3
3 4 x β 2 > 4 3 4 x β 2 > 4 or 4 ( 2 β x ) > 0 4 ( 2 β x ) > 0
2 3 x β 3 > 5 2 3 x β 3 > 5 or 3 ( 5 β x ) > 6 3 ( 5 β x ) > 6
3 x β 2 > 4 3 x β 2 > 4 or 5 x β 3 β€ 7 5 x β 3 β€ 7
2 ( x + 3 ) β₯ 0 2 ( x + 3 ) β₯ 0 or 3 ( x + 4 ) β€ 6 3 ( x + 4 ) β€ 6
1 2 x β 3 β€ 4 1 2 x β 3 β€ 4 or 1 3 ( x β 6 ) β₯ β2 1 3 ( x β 6 ) β₯ β2
3 4 x + 2 β€ β1 3 4 x + 2 β€ β1 or 1 2 ( x + 8 ) β₯ β3 1 2 ( x + 8 ) β₯ β3
Mixed practice
3 x + 7 β€ 1 3 x + 7 β€ 1 and 2 x + 3 β₯ β5 2 x + 3 β₯ β5
6 ( 2 x β 1 ) > 6 6 ( 2 x β 1 ) > 6 and 5 ( x + 2 ) β₯ 0 5 ( x + 2 ) β₯ 0
4 β 7 x β₯ β3 4 β 7 x β₯ β3 or 5 ( x β 3 ) + 8 > 3 5 ( x β 3 ) + 8 > 3
1 2 x β 5 β€ 3 1 2 x β 5 β€ 3 or 1 4 ( x β 8 ) β₯ β3 1 4 ( x β 8 ) β₯ β3
β5 β€ 2 x β 1 < 7 β5 β€ 2 x β 1 < 7
1 5 ( x β 5 ) + 6 < 4 1 5 ( x β 5 ) + 6 < 4 and 3 β 2 3 x < 5 3 β 2 3 x < 5
4 x β 2 > 6 4 x β 2 > 6 or 3 x β 1 β€ β2 3 x β 1 β€ β2
6 x β 3 β€ 1 6 x β 3 β€ 1 and 5 x β 1 > β6 5 x β 1 > β6
β2 ( 3 x β 4 ) β€ 2 β2 ( 3 x β 4 ) β€ 2 and β4 ( x β 1 ) < 2 β4 ( x β 1 ) < 2
β5 β€ 3 x β 2 β€ 4 β5 β€ 3 x β 2 β€ 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 personβs 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.
β 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.
β 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]
[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]
[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]
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]
[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]
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]
[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]
Contribute!
Licenses & attributions, cc licensed content, original.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution .
CC licensed content, Shared previously
- 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 β€ -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 (βͺ) 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 β€ 3x + 2 < 5,” the two inequalities are “-4 β€ 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 β€ 0 OR 3y β₯ 0.”
Solve each inequality separately: “2y – 2 β€ 0” becomes “2y β€ 2,” then “y β€ 1.” “3y β₯ 0” implies “y β₯ 0.”
Combine the solutions using “OR”: “y β€ 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 (β©) combines solutions for “AND,” showing common values that satisfy both inequalities. Union (βͺ) 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 β€ 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 βor,β and express solutions both graphically and with interval notation
- Solve compound inequalities involving βand,β 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 βorβ
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]3xβ1<8[/latex] or [latex]xβ5>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.
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]β3yβ2\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]β4[/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]
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]5zβ3\gtβ18[/latex] or [latex]β2zβ1\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.
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.
Inequality: [latex] \displaystyle x\ge 4[/latex]
Interval: [latex]\left[4,\infty\right)[/latex]
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]
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]
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.
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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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.
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.
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.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
Have an account?
Tally Charts and Graphs
Graphing inequalities, interval notation review.
Solving Compound Inequalities
Mathematics.
14 questions
- 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 β€ 3 1 β€ x β€ 7 1 < x β€ 3 -2/3 < x β€ 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 β€ 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
IMAGES
VIDEO
COMMENTS
SOLVING COMPOUND INEQUALITIES 4.6 (34 reviews) compound inequality Click the card to flip π a statement formed by two or more inequalities Click the card to flip π 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
Solving Compound Inequalities Flashcards | Quizlet Solving Compound Inequalities -10 < x < 8 Click the card to flip π Solve the Inequality. -1 < 9 + x < 17 Click the card to flip π 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
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 ...
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
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
Study with Quizlet and memorize flashcards containing terms like 3<x-1<7, 4<2x+2<8, 10<-2x<22 and more.
Solving Compound Inequalities! 5 < x < 7 Click the card to flip π 8 < x + 3 < 10 Click the card to flip π 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
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.
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.
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
You might need: Calculator Solve for x . β 7 x β 50 β€ β 1 AND β 6 x + 70 > β 2 Choose 1 answer: x β₯ β 7 A x β₯ β 7 β 7 β€ x < 12 B β 7 β€ 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.
Write the solution in interval notation. [ β 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 β 7 < 9 and 5 x + 8 β₯ 3. Answer.
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.
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.
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.
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, β) Graph: [/hidden-answer] Example. Solve for x: 5 x β 2 β€ 3 and 4 x + 7 > 3.
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 ...
What is a Compound Inequality? Start by solving each inequality separately: -2 < 2x - 3β³ becomes "1 < 2x. 2x - 3 < 5β³ becomes "2x < 8. Then, divide each inequality by 2: Solve each inequality separately: Solve each inequality separately: "2y - 2 β€ 0" becomes "2y β€ 2," then "y β€ 1." "3y β₯ 0" implies "y β₯ 0." About us Contact us
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.
Solve the compound inequality. Graph the solution and write the solution in interval notation: 1 5x β 5 β₯ β 3 and β 4(x β 1) β₯ β 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.
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.
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"βwe 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 ...
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