# 6.6: Solutions to Compound Inequalities

**At Grade**Created by: CK-12

**Practice**Solutions to Compound Inequalities

What if you had a compound inequality like

### Solutions to Compound Inequalities

When we solve compound inequalities, we separate the inequalities and solve each of them separately. Then, we combine the solutions at the end.

#### Solve the following compound inequalities and graph the solution set.

a)

First we re-write the compound inequality as two separate inequalities with *and*. Then solve each inequality separately.

b)

Re-write the compound inequality as two separate inequalities with *and*. Then solve each inequality separately.

#### Solve the following compound inequalities and graph the solution set.

a)

Solve each inequality separately:

b)

Solve each inequality separately:

One thing you may notice in the video for this Concept is that in the second problem, the two solutions joined with “or” overlap, and so the solution ends up being the set of all real numbers, or

Compound inequalities with “and,” meanwhile, can turn out to have *no* solutions. For example, the inequality “

**Solve Real-World Problems Using Compound Inequalities**

Many application problems require the use of compound inequalities to find the solution.

#### Solve the Real-World Problem

*The speed of a golf ball in the air is given by the formula v=−32t+80. When is the ball traveling between 20 ft/sec and 30 ft/sec?*

First we set up the inequality

Then we separate the compound inequality and solve each separate inequality:

To check the answer, we plug in the minimum and maximum values of

For

For

So the speed is between 20 and 30 ft/sec. **The answer checks out.**

### Example

*William’s pick-up truck gets between 18 to 22 miles per gallon of gasoline. His gas tank can hold 15 gallons of gasoline. If he drives at an average speed of 40 miles per hour, how much driving time does he get on a full tank of gas?*

Let

Since the truck gets between 18 and 22 miles/gallon, we set up the compound inequality

**Andrew can drive between 6.75 and 8.25 hours on a full tank of gas.**

If we plug in

If we plug in

**The answer checks out.**

### Review

Solve the following compound inequalities and graph the solution on a number line.

−5≤x−4≤13 1≤3x+5≤4 - \begin{align*}-12 \le 2-5x \le 7\end{align*}
- \begin{align*}\frac{3}{4} \le 2x+9 \le \frac{3}{2}\end{align*}
- \begin{align*}-2 \le \frac{2x-1}{3} < -1\end{align*}
- \begin{align*}4x-1 \ge 7\end{align*} or \begin{align*}\frac{9x}{2} < 3\end{align*}
- \begin{align*}3-x < -4\end{align*} or \begin{align*}3-x > 10\end{align*}
- \begin{align*}\frac{2x+3}{4} < 2\end{align*} or \begin{align*}-\frac{x}{5} + 3 < \frac{2}{5}\end{align*}
- \begin{align*}2x-7 \le -3\end{align*} or \begin{align*}2x-3 > 11\end{align*}
- \begin{align*}4x+3< 9\end{align*} or \begin{align*}-5x+4 \le -12\end{align*}
- How would you express the answer to problem 1 as a set?
- How would you express the answer to problem 1 as an interval?
- How would you express the answer to problem 6 as a set?
- Could you express the answer to problem 6 as a single interval? Why or why not?
- How would you express the first part of the solution in interval form?
- How would you express the second part of the solution in interval form?

- Express the answers to problems 2 through 5 in interval notation.
- Solve the inequality “\begin{align*}x \ge -3\end{align*} or \begin{align*}x < 1\end{align*}” and express the answer in interval notation.
- How many solutions does the inequality “\begin{align*}x \ge 2\end{align*} and \begin{align*}x \le 2\end{align*}” have?
- To get a grade of B in her Algebra class, Stacey must have an average grade greater than or equal to 80 and less than 90. She received the grades of 92, 78, 85 on her first three tests.
- Between which scores must her grade on the final test fall if she is to receive a grade of B for the class? (Assume all four tests are weighted the same.)
- What range of scores on the final test would give her an overall grade of C, if a C grade requires an average score greater than or equal to 70 and less than 80?
- If an A grade requires a score of at least 90, and the maximum score on a single test is 100, is it possible for her to get an A in this class? (Hint: look again at your answer to part a)

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.6.

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### Image Attributions

Here you'll learn how to separate compound inequalities with *"and"* or *"or"* and solve them separately. You'll then learn how to combine your answers into a single solution and graph the solution set.

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