Lesson 16
Interpreting Inequalities
16.1: Solve Some Inequalities! (5 minutes)
Warmup
This warmup is an opportunity for students to recall understandings and techniques from the previous lesson.
Launch
Optionally, provide access to blank number lines.
Student Facing
For each inequality, find the value or values of \(x\) that make it true.

\(8x+21 \leq 56\)

\(56 < 7(7x)\)
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students express the solution in words or by graphing on a number line, applaud their use of these representations. Encourage them to attempt to express the solution using the efficient notation, as well. Direct their attention to any anchor charts or notes that remind them of the meaning of the symbols involved.
Activity Synthesis
Ask one student to share their process for reasoning about a solution to each problem. Address and resolve any discrepancies that arise.
16.2: Club Activities Matching (10 minutes)
Activity
In this activity, students analyze four situations and select the inequality that best represents the situation. (In the activity that follows, students will work in small groups to create a visual display showing the solution for one of these situations.)
Launch
Tell students that their job in this activity is to read four situations carefully and decide which inequality best represents the situation. In the next activity, they will be responsible for writing a solution for one of these situations. Give 5–10 minutes of quiet work time.
Supports accessibility for: Visualspatial processing
Design Principle(s): Support sensemaking; Maximize metaawareness
Student Facing
Choose the inequality that best matches each given situation. Explain your reasoning.
 The Garden Club is planting fruit trees in their school’s garden. There is one large tree that needs 5 pounds of fertilizer. The rest are newly planted trees that need \(\frac12\) pound fertilizer each.
 \(25x + 5 \leq \frac12\)
 \(\frac12 x + 5 \leq 25\)
 \(\frac12 x + 25 \leq 5\)
 \(5x + \frac12 \leq 25\)
 The Chemistry Club is experimenting with different mixtures of water with a certain chemical (sodium polyacrylate) to make fake snow.
To make each mixture, the students start with some amount of water, and then add \(\frac17\) of that amount of the chemical, and then 9 more grams of the chemical. The chemical is expensive, so there can’t be more than a certain number of grams of the chemical in any one mixture. \(\frac17 x + 9 \leq 26.25\)
 \(9x + \frac17 \leq 26.25\)
 \(26.25x + 9 \leq \frac17\)

\(\frac17 x + 26.25 \leq 9\)
 The Hiking Club is on a hike down a cliff. They begin at an elevation of 12 feet and descend at the rate of 3 feet per minute.
 \(37x  3 \geq 12\)
 \(3x 37 \geq 12\)
 \(12  3x \geq \text37\)

\(12x  37 \geq \text3\)
 The Science Club is researching boiling points. They learn that at high altitudes, water boils at lower temperatures. At sea level, water boils at \(212^\circ \text{F}\). With each increase of 500 feet in elevation, the boiling point of water is lowered by about \(1^\circ \text{F}\).
 \(212  \frac{1}{500}e < 195\)
 \(\frac{1}{500}e  195 < 212\)
 \(195  212e < \frac{1}{500}\)
 \(212  195e < \frac{1}{500}\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
At this time, consider not validating which inequalities are correct. When students get into groups for the next activity, they can compare their responses with the members of their groups and resolve any discrepancies at that time.
16.3: Club Activities Display (20 minutes)
Activity
In this activity, students interpret parts of an inequality in context, term by term; for example, what quantity must \(\frac{1}{2}x\) represent? Then they make sense of the entire inequality by thinking about what question would be answered by the solution to the inequality. Notice groups that create displays that communicate their mathematical thinking clearly, contain an error that would be instructive to discuss, or organize the information in a way that is useful for all to see. At this point, there is very little scaffolding for the solving of the inequality itself.
Launch
Arrange students in groups of 2–3 and provide tools for making a visual display. Assign one situation to each group. Note that the level of difficulty increases for the situations, so this is an opportunity to differentiate by assigning more or less challenging situations to different groups.
Supports accessibility for: Attention; Socialemotional skills
Student Facing
Your teacher will assign your group one of the situations from the last task. Create a visual display about your situation. In your display:
 Explain what the variable and each part of the inequality represent
 Write a question that can be answered by the solution to the inequality
 Show how you solved the inequality

Explain what the solution means in terms of the situation
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
\(\{3,4,5,6\}\) is a set of four consecutive integers whose sum is 18.
 How many sets of three consecutive integers are there whose sum is between 51 and 60? Can you be sure you’ve found them all? Explain or show your reasoning.
 How many sets of four consecutive integers are there whose sum is between 59 and 82? Can you be sure you’ve found them all? Explain or show your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Select groups to share their visual displays. Encourage students to ask questions about the mathematical thinking or design approach that went into creating the display. Here are questions for discussion, if not already mentioned by students:
 How did you figure out what the \(\frac{x}{7}\) term represents?
 How did you decide on the direction of the inequality for the solutions?
 Did anyone with the same problem do one of the steps differently? Share what you did differently so we can learn from what happened.
 How do you know there are 25 pounds of fertilizer available?
Alternatively, have students do a “gallery walk” in which they leave written feedback on sticky notes for the other groups. Here is guidance for the kind of feedback students should aim to give each other:
 What is one thing that group did that would have made your project better if you had done it?
 What is one thing your group did that would have improved their project if they did it too?
 How did the group decide the direction of inequality for the solutions?
 Does their answer make sense in the situation?
 Is their mathematics clear and correct?
 If there was a mistake, what could they be more careful about in similar problems?
Design Principle(s): Maximize metaawareness; Support sensemaking
Lesson Synthesis
Lesson Synthesis
In this lesson, we saw how inequalities can be applied to realworld situations. Some questions to bring this work together:
 Suppose your friend asks you to write some practice problems for solving inequalities. You want to write an inequality that has a solution of \(x\leq\text8\frac23\). Describe how to write such an inequality.
 Think about an afterschool activity in which you are involved. Write an inequality that represents a situation related to that activity. Be prepared to share the inequality and an explanation of its terms with the class.
If time allows, have students solve their inequalities.
16.4: Cooldown  Party Decorations (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
We can represent and solve many realworld problems with inequalities. Writing the inequalities is very similar to writing equations to represent a situation. The expressions that make up the inequalities are the same as the ones we have seen in earlier lessons for equations. For inequalities, we also have to think about how expressions compare to each other, which one is bigger, and which one is smaller. Can they also be equal?
For example, a school fundraiser has a minimum target of \$500. Faculty have donated \$100 and there are 12 student clubs that are participating with different activities. How much money should each club raise to meet the fundraising goal? If \(n\) is the amount of money that each club raises, then the solution to \(100+12n=500\) is the minimum amount each club has to raise to meet the goal. It is more realistic, though, to use the inequality \(100+12n\geq500\) since the more money we raise, the more successful the fundraiser will be. There are many solutions because there are many different amounts of money the clubs could raise that would get us above our minimum goal of \$500.