Lesson 15Efficiently Solving Inequalities

Learning Goal

Let’s solve more complicated inequalities.

Learning Targets

I can graph the solutions to an inequality on a number line.
I can solve inequalities by solving a related equation and then checking which values are solutions to the original inequality.

Lesson Terms

solution to an inequality

Warm Up: Lots of Negatives

Here is an inequality: $- x \geq - 4$ .

Predict what you think the solutions on the number line will look like.
Select all the values that are solutions to $- x \geq - 4$ :
1. 3
2. -3
3. 4
4. -4
5. 4.001
6. -4.001
Graph the solutions to the inequality on the number line:

Activity 1: Inequalities with Tables

Problem 1

Let’s investigate the inequality $x - 3 > - 2$ .

Complete the table.
$x$
$- 4$
$- 3$
$- 2$
$- 1$
$0$
$1$
$2$
$3$
$4$
$x - 3$
$- 7$
$- 5$
$- 1$
$1$
For which values of $x$ is it true that $x - 3 = - 2$ ?
For which values of $x$ is it true that $x - 3 > - 2$ ?
Graph the solutions to $x - 3 > - 2$ on the number line:

$x$	$- 4$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$	$4$
$x - 3$	$- 7$		$- 5$				$- 1$		$1$

Problem 2

Here is an inequality: $2 x < 6$ .

Predict which values of $x$ will make the inequality $2 x < 6$ true.
Complete the table. Does it match your prediction?
$x$
$- 4$
$- 3$
$- 2$
$- 1$
$0$
$1$
$2$
$3$
$4$
$2 x$
Graph the solutions to $2 x < 6$ on the number line:

$x$	$- 4$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$	$4$
$2 x$

Problem 3

Here is an inequality: $- 2 x < 6$ .

Predict which values of $x$ will make the inequality $- 2 x < 6$ true.
Complete the table. Does it match your prediction?
$x$
$- 4$
$- 3$
$- 2$
$- 1$
$0$
$1$
$2$
$3$
$4$
$- 2 x$
Graph the solutions to $- 2 x < 6$ on the number line:
How are the solutions to $2 x < 6$ different from the solutions to $- 2 x < 6$ ?

$x$	$- 4$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$	$4$
$- 2 x$

Activity 2: Which Side are the Solutions?

Problem 1

Let’s investigate $- 4 x + 5 \geq 25$ .

Solve $- 4 x + 5 = 25$ .
Is $- 4 x + 5 \geq 25$ true when $x$ is $0$ ? What about when $x$ is $7$ ? What about when $x$ is $- 7$ ?
Graph the solutions to $- 4 x + 5 \geq 25$ on the number line.

Problem 2

Let’s investigate $\frac{4}{3} x + 3 < \frac{23}{3}$ .

Solve $\frac{4}{3} x + 3 = \frac{23}{3}$ .
Is $\frac{4}{3} x + 3 < \frac{23}{3}$ true when $x$ is 0?
Graph the solutions to $\frac{4}{3} x + 3 < \frac{23}{3}$ on the number line.

Problem 3

Solve the inequality $3 (x + 4) > 17.4$ and graph the solutions on the number line.

a blank number line with 11 evenly spaced tick marks

Problem 4

Solve the inequality $- 3 (x - \frac{4}{3}) \leq 6$ and graph the solutions on the number line.

Are you ready for more?

Write at least three different inequalities whose solution is $x > - 10$ . Find one with $x$ on the left side that uses a $<$ .

Lesson Summary

Here is an inequality: $3 (10 - 2 x) < 18$ . The solution to this inequality is all the values you could use in place of $x$ to make the inequality true.

In order to solve this, we can first solve the related equation $3 (10 - 2 x) = 18$ to get the solution $x = 2$ . That means $2$ is the boundary between values of $x$ that make the inequality true and values that make the inequality false.

To solve the inequality, we can check numbers greater than $2$ and less than $2$ and see which ones make the inequality true.

Let’s check a number that is greater than $2$ : $x = 5$ . Replacing $x$ with $5$ in the inequality, we get $3 (10 - 2 \cdot 5) < 18$ or just $0 < 18$ . This is true, so $x = 5$ is a solution. This means that all values greater than $2$ make the inequality true.

We can write the solutions as $x > 2$ and also represent the solutions on a number line:

A number line from -3 to 5 with a open circle at 2 and an arrow extending right to 5

Notice that 2 itself is not a solution because it’s the value of $x$ that makes $3 (10 - 2 x)$ equal to $18$ , and so it does not make $3 (10 - 2 x) < 18$ true.

For confirmation that we found the correct solution, we can also test a value that is less than $2$ . If we test $x = 0$ , we get $3 (10 - 2 \cdot 0) < 18$ or just $30 < 18$ . This is false, so $x = 0$ and all values of $x$ that are less than $2$ are not solutions.