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Lesson 15Efficiently Solving Inequalities

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.

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:
    A blank 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:
      A blank 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:

      A blank 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:
      A blank 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.
      A blank 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.

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

Are you ready for more?

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

Lesson 15 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:

An inequality is 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.

Lesson 15 Practice Problems

    1. Consider the inequality \text-1 \leq \frac{x}{2} .
      1. Predict which values of x will make the inequality true.
      2. Complete the table to check your prediction.
        x -4 -3 -2 -1 0 1 2 3 4
        \frac{x}{2}
    2. Consider the inequality 1 \leq \text-\frac{x}{2} .
      1. Predict which values of x will make it true.
      2. Complete the table to check your prediction.
        x -4 -3 -2 -1 0 1 2 3 4
        \text-\frac{x}{2}

  2. Diego is solving the inequality 100-3x \ge \text-50 . He solves the equation 100-3x = \text-50 and gets x=50 . What is the solution to the inequality?

    1. x < 50
    2. x \le 50
    3. x > 50
    4. x \ge 50
  3. Solve the inequality \text-5(x-1)>\text-40 , and graph the solution on a number line.

  4. Select all values of x that make the inequality \text-x+6\ge10 true.

    1. -3.9
    2. 4
    3. -4.01
    4. -4
    5. 4.01
    6. 3.9
    7. 0
    8. -7

  5. Draw the solution set for each of the following inequalities.

    1. x>7

      A number line with the numbers negative 10 through 9 indicated.

    2. x\geq\text-4.2

      A number line with the numbers negative 10 through 9 indicated.

  6. The price of a pair of earrings is $22 but Priya buys them on sale for $13.20.

    1. By how much was the price discounted?
    2. What was the percentage of the discount?