# Lesson 15

Efficiently Solving Inequalities

Let’s solve more complicated inequalities.

### 15.1: Lots of Negatives

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

1. Predict what you think the solutions on the number line will look like.

2. Select all the values that are solutions to $$\text-x \geq \text-4$$:
1. 3
2. -3
3. 4
4. -4
5. 4.001
6. -4.001
3. Graph the solutions to the inequality on the number line:

### 15.2: Inequalities with Tables

1. Let's investigate the inequality $$x-3>\text-2$$.

 $$x$$ $$x-3$$ -4 -3 -2 -1 0 1 2 3 4 -7 -5 -1 1
1. Complete the table.
2. For which values of $$x$$ is it true that $$x - 3 = \text-2$$?
3. For which values of $$x$$ is it true that $$x - 3 > \text-2$$?
4. Graph the solutions to $$x - 3 > \text-2$$ on the number line:
2. Here is an inequality: $$2x<6$$.

1. Predict which values of $$x$$ will make the inequality $$2x < 6$$ true.
2. Complete the table. Does it match your prediction?

 $$x$$ $$2x$$ -4 -3 -2 -1 0 1 2 3 4
3. Graph the solutions to $$2x < 6$$ on the number line:

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

1. Predict which values of $$x$$ will make the inequality $$\text-2x < 6$$ true.
2. Complete the table. Does it match your prediction?

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

3. Graph the solutions to $$\text-2x < 6$$ on the number line:
4. How are the solutions to $$2x<6$$ different from the solutions to $$\text-2x<6$$?

### 15.3: Which Side are the Solutions?

1. Let’s investigate $$\text-4x + 5 \geq 25$$.
1. Solve $$\text-4x+5 = 25$$.
2. Is $$\text-4x + 5 \geq 25$$ true when $$x$$ is 0? What about when $$x$$ is 7? What about when $$x$$ is -7?
3. Graph the solutions to $$\text-4x + 5 \geq 25$$ on the number line.
2. Let's investigate $$\frac{4}{3}x+3 < \frac{23}{3}$$.
1. Solve $$\frac43x+3 = \frac{23}{3}$$.
2. Is $$\frac{4}{3}x+3 < \frac{23}{3}$$ true when $$x$$ is 0?
3. Graph the solutions to $$\frac{4}{3}x+3 < \frac{23}{3}$$ on the number line.

3. Solve the inequality $$3(x+4) > 17.4$$ and graph the solutions on the number line.
4. Solve the inequality $$\text-3\left(x-\frac43\right) \leq 6$$ and graph the solutions on the number line.

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

### Summary

Here is an inequality: $$3(10-2x) < 18$$. 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 $$3(10-2x) = 18$$ 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 $$3(10-2 \boldcdot 5) < 18$$ 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:

Notice that 2 itself is not a solution because it's the value of $$x$$ that makes $$3(10-2x)$$ ​equal to 18, and so it does not make $$3(10-2x) < 18$$ 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 $$3(10-2 \boldcdot 0) < 18$$ or just $$30 < 18$$. This is false, so $$x = 0$$ and all values of $$x$$ that are less than 2 are not solutions.

### Glossary Entries

• solution to an inequality

A solution to an inequality is a number that can be used in place of the variable to make the inequality true.

For example, 5 is a solution to the inequality $$c<10$$, because it is true that $$5<10$$. Some other solutions to this inequality are 9.9, 0, and -4.