Modeling with Inequalities

Playlist Timing

5 min

Problem

Elena is trying to create a playlist that lasts no more than 2 hours (120 minutes). She has already added songs that total 15 minutes. She reads that the average song length on her music streaming service is 3.5 minutes. Elena writes the inequality 3.5x+15≥1203.5x + 15 \geq 120 and solves it to find the solution x≥30x \geq 30.

  1. Explain how you know Elena made a mistake based on her solution.
  2. Fix Elena’s inequality and explain what each part of the inequality means.

Answer

  1. Sample response: x≥30x \geq 30 means Elena can add more than 30 songs on the playlist. This doesn’t make sense because there should be a maximum limit on songs rather than a minimum limit.
  2. The correct inequality is 3.5x+15≤1203.5x+15 \leq 120. The number 3.5 represents the average length of each song. The variable xx represents the number of songs that Elena adds. The 15 represents the 15 minutes of songs that are already on the playlist. The ≤120\leq 120 represents that the total number of minutes has to be less than or equal to 120.

Sample Response

  1. Sample response: x≥30x \geq 30 means Elena can add more than 30 songs on the playlist. This doesn’t make sense because there should be a maximum limit on songs rather than a minimum limit.
  2. The correct inequality is 3.5x+15≤1203.5x+15 \leq 120. The number 3.5 represents the average length of each song. The variable xx represents the number of songs that Elena adds. The 15 represents the 15 minutes of songs that are already on the playlist. The ≤120\leq 120 represents that the total number of minutes has to be less than or equal to 120.
Responding to Student Thinking
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Response: By this point in the unit, there should be some student mastery of interpreting and solving inequalities. If students struggle, plan to make time to revisit related work in the section referred to here. See the Course Guide for ideas to help students re-engage with earlier work.

Standards
Addressing
  • 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. \$