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 $x-3>\text-2$.

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

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

    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?

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

    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.

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.

Practice Problems ▶