Topics
1. Equations & Inequalities in One Variable
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United States of AmericaFL
Grade 912

1.07 Compound inequalities

Lesson
Practice
Lesson

Concept summary

A compound inequality is a conjunction of two or more inequalities. The set of solutions for a compound inequality are the values which make all of the inequalities true.

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  • We use "and" to indicate that a value must satisfy both inequalities in order to be in the solution set. For example:x \lt 3 \text{ and } x \geq -2We can also write this compound inequality more simply as -2\leq x <3
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  • We use "or" to indicate that a value need only satisfy at least one inequality in order to be in the solution set. For example: x \gt 3 \text{ or } x \leq -2

Worked examples

Example 1

Write a compound inequality to represent the solution set shown on the number line.

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Approach

We want to describe the compound inequality then write the solution to the compound inequality algebraically.

Description: x is less than -1 or x is greater than or equal to 2

Solution

Compound inequality: x < -1 or x \geq 2

Example 2

Find the solution set of the compound inequality x \geq -5 and x < 3 using set-builder notation and plot the solution on a number line.

Approach

  1. Plot x \geq -5 on a number line.

  2. Plot x < 3 on a number line.

  3. Find the solution of the compound inequality by comparing the number lines.

  4. Represent the solution graphically.

  5. Represent the solution using set-builder notation.

Solution

Plotting x \geq -5:

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The minimum value, -5, is included in the interval and is represented by a filled circle.

Plotting x< 3:

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The maximum value, 3, is not included in the interval and is represented by a unfilled circle.

Finding the solution to the compound inequality:

We want the solution to both x \geq -5 and x \lt 3. The solution to the compound inequality will be the values greater than or equal to -5 and less than 3.

Solution represented graphically on a number line:

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Solution represented algebraically using set-builder notation: \left\{x\in\Reals\vert -5\leq x<3\right\}

Reflection

We can check our solution(s) are correct by taking a number in our solution set and seeing if it works in both inequalities.

For example, we can see that 0 falls within the region shown on the number line, and \\-5\leq 0 < 3 as required.

Example 3

The formula for converting temperatures from Celsius to Fahrenheit is: F = \dfrac{9}{5} C+32

During a recent year, the average temperatures in Tampa, Florida ranged from 59 \degree to 95 \degree Fahrenheit.

Write a compound inequality to solve for the corresponding range of values of C, the temperature in Florida in degrees Celsius.

Approach

The degrees in Fahrenheit must be between 59 and 95 inclusive, so the minimum is 59 and the maximum is 95. The middle part of our compound inequality is going to be the formula for converting temperatures.

Once we have the compound inequality we can solve it.

Solution

\displaystyle 59\displaystyle \leq\displaystyle \frac{9}{5} C+32 \leq 95
\displaystyle 27\displaystyle \leq\displaystyle \frac{9}{5}C \leq 63Subtract 32 from each section of the inequality
\displaystyle 135\displaystyle \leq\displaystyle 9C \leq 315Multiply each section of the inequality by 5
\displaystyle 15\displaystyle \leq\displaystyle C \leq 35Divide each section of the inequality by 9

The average temperatures in Tampa ranged from 15 \degree to 35 \degree Celsius.

Reflection

When solving a compound inequality, whatever is done to one part of the inequality must be done to all parts.

Outcomes

MA.912.AR.2.6

Given a mathematical or real-world context, write and solve one-variable linear inequalities, including compound inequalities. Represent solutions algebraically or graphically.

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