Efficiently Solving Inequalities

5 min

Teacher Prep
Setup
3 minutes quiet work time followed by whole-class discussion.

Narrative

In this activity, students predict what the solutions to -x≥-4\text- x \ge \text- 4 will look like, then test values, and then graph solutions.

Do not formalize a procedure for “flipping the inequality” when multiplying or dividing by a negative. Monitor for students who predict solution sets that are incorrect because of the sign.

Launch

Give students 3 minutes of quiet work time followed by a whole-class discussion. Students are not expected to be able to make correct predictions in the first question. The purpose of this question is to prompt students to think about how negative values in an inequality affect the solution. It also emphasizes that we should not jump to conclusions about solutions without carefully studying what the inequality means.

Student Task

Here is an inequality: -x≥-4\text-x \geq \text-4

  1. Without testing any values, predict what the solutions to this inequality will look like on the number line.
    • Where will the starting (or boundary) point be?
    • Will the circle at that point be filled or unfilled?
    • Will the shading go to the right or to the left of that point?
  2. Test each value to see whether it is a solution to the inequality -x≥-4\text-x \geq \text-4.

    1. 3

    2. -3

    3. 4

    4. -4

    5. 4.001

    6. -4.001

  3. Graph all possible solutions to the inequality on the number line:

    Blank number line, 11 evenly spaced tick marks, no labels.

Sample Response

  1. Sample response: The starting point will be 4 and its circle will be filled in. The shading will go to the left of 4. 
  2.  
    1. yes
    2. yes
    3. yes
    4. yes
    5. no
    6. yes

  3. Sample response:

    <p>A number line with the numbers negative 5 through 5 indicated. An filled-in circle is indicated at 4 and an arrow is drawn from the circle extending to the left.</p>

Activity Synthesis (Teacher Notes)

The purpose of the discussion is to highlight how negatives in the inequality sometimes make it hard to predict what the solutions will be.

Select students to share how their predictions differed from their final solutions. 

To illustrate a simple case where solutions go in the opposite direction on the number line, ask how the solutions to -x≥-4\text{-}x \geq \text{-}4 are different from the solutions to x≥4x \geq 4. 

Standards
Building Toward
  • 7.EE.4.b·Solve word problems leading to inequalities of the form px + q &gt; r or px + q &lt; r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. <em>For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.</em>
  • 7.EE.B.4.b·Solve word problems leading to inequalities of the form <span class="math">\(px + q &gt; r\)</span> or <span class="math">\(px + q &lt; r\)</span>, where <span class="math">\(p\)</span>, <span class="math">\(q\)</span>, and <span class="math">\(r\)</span> are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. \$

15 min

15 min