Finding Solutions to Inequalities in Context

10 min

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

Narrative

In this activity, students solve some equations and some related inequalities. This Warm-up highlights the link between an inequality and its associated equation. Monitor for students who use the value -10 as a boundary as they test values to find solutions to the inequalities.

Launch

Give students 5 minutes of quiet work time followed by a whole-class discussion. Optionally, provide students with blank number lines for scratch work.

Student Task

  1. Solve -x=10\text-x = 10
  2. Find 2 values of xx that make this inequality true: -x>10\text-x > 10
  3. Solve 2x= -202x = \text-20
  4. Find 2 values of xx that make this inequality true: 2x>-202x > \text-20

Sample Response

  1. -10
  2. Sample responses: -12, -28.7, -209 (Any value that is less than -10 works.)
  3. -10
  4. Sample responses: -9, 0, 823482\frac34 (Any value that is greater than -10 works.) 
Activity Synthesis (Teacher Notes)

The purpose of this discussion is for students to understand the term solution to an inequality and for them to recognize that the number that makes the two sides of an inequality equal is a good region of the number line to start looking for solutions.

Display two number lines for all to see that each include -10 and some integral values to its left and right. Ask a few students to share their responses to the first two questions, recording their responses on one number line and gauging the class for agreement. Ask a few students to share their responses to the last two questions, recording their responses on the other number line and gauging the class for agreement. 

Highlight the fact that -x=10\text{-}x = 10 and 2x=-202x = \text{-}20 have the same solution (-10), but the inequalities -x>10\text{-}x > 10 and 2x>-202x > \text{-}20 don't have the same solutions. Ask:

  • “How do you know when a value is a solution to an equation?”
  • “How do you know when a value is a solution to an inequality?”
  • “How are solutions to an inequality different from solutions to an equation?”

Select students to share strategies they had for finding solutions to the inequalities in this activity. If not mentioned by students, discuss the fact that since -10 makes the two sides of the inequality equal, the region of values around -10 is a good place to start looking for solutions.

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