Lesson 4

Finding Solutions to Inequalities in Context

4.1: Solutions to Equations and Solutions to Inequalities (10 minutes)

Warm-up

This warm-up highlights the link between an inequality and its associated equation. This will be solidified throughout the lesson as students solve the associated equation and reason in context to determine the direction of inequality. Notice students who use the value -10 as a boundary as they test values to find solutions to the inequalities.

Teacher Notes for IM 6–8 Math Accelerated
Adjust this activity to 5 minutes.

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 Facing

1. Solve $$\text-x = 10$$

2. Find 2 solutions to $$\text-x > 10$$

3. Solve $$2x = \text-20$$

4. Find 2 solutions to $$2x > \text-20$$

Activity Synthesis

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 $$\text-x = 10$$ and $$2x = \text-20$$ have the same solution (-10), but the inequalities $$\text-x > 10$$ and $$2x > \text-20$$ don't have the same solutions. Select students to share strategies they had for finding solutions. If not mentioned by students, discuss the fact that since -10 makes the sides equal, the neighborhood of values around -10 is a good place to start looking for solutions.

4.2: Earning Money for Soccer Stuff (15 minutes)

Activity

Previously in this unit, students wrote expressions and equations that are similar to the ones in this activity. Here, they are prompted in a scaffolded way to notice that they can express not just that an outcome can be equal to a value, but that an outcome can be at least as much as a value by using the new notation $$\geq$$.

Teacher Notes for IM 6–8 Math Accelerated
Students were introduced to the symbols $$\leq$$ and $$\geq$$ and their meaning at the start of this unit. This activity allows a more formal introduction to their use. At the start of the launch, ask students, "Elena spent less than $40 on presents for her family. Han spent no more than$40 for presents for his family. Would the inequalities for these situations be the same or different?" (They would be different, because Elena could not have spent $40, so $$E<40$$, while Han could have spent$40, so $$H\leq40$$.) Select students to share their responses. If not brought up by students and time allows, display a number line for Elena and Han and highlight the open circle used for Elena, since $40 is not part of her answer, and the closed circle used for Han, since$40 is part of his answer.

Launch

Optionally, provide access to blank number lines to use for scratch work.

Arrange students in groups of 2. Allow 10 minutes of quiet work time and partner discussion followed by a whole-class discussion. Depending on the needs of your class, you may decide to ask students to pause after the first question for the whole-class discussion before tackling the second question.

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who correctly write an expression but struggle in writing an equation with the correct inequality sign. Consider pausing for a brief whole-class discussion inviting students to share strategies for determining the correct sign.
Supports accessibility for: Memory; Organization
Conversing: MLR5 Co-craft Questions. Reveal only the context for Andre’s summer job, without revealing the questions that follow. Ask students to create mathematical questions about this situation. Give students 1–2 minutes to write down mathematical questions that could be asked about the situation. Invite students to share their questions with the class, before revealing the rest of the activity. Listen for an amplify questions that contain the phrase “at least.”
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

1. Andre has a summer job selling magazine subscriptions. He earns $25 per week plus$3 for every subscription he sells. Andre hopes to make at least enough money this week to buy a new pair of soccer cleats.

1. Let $$n$$ represent the number of magazine subscriptions Andre sells this week. Write an expression for the amount of money he makes this week.
2. The least expensive pair of cleats Andre wants costs $68. Write and solve an equation to find out how many magazine subscriptions Andre needs to sell to buy the cleats. 3. If Andre sold 16 magazine subscriptions this week, would he reach his goal? Explain your reasoning. 4. What are some other numbers of magazine subscriptions Andre could have sold and still reached his goal? 5. Write an inequality expressing that Andre wants to make at least$68.
6. Write an inequality to describe the number of subscriptions Andre must sell to reach his goal.
2. Diego has budgeted $35 from his summer job earnings to buy shorts and socks for soccer. He needs 5 pairs of socks and a pair of shorts. The socks cost different amounts in different stores. The shorts he wants cost$19.95.

1. Let $$x$$ represent the price of one pair of socks. Write an expression for the total cost of the socks and shorts.
2. Write and solve an equation that says that Diego spent exactly $35 on the socks and shorts. 3. List some other possible prices for the socks that would still allow Diego to stay within his budget. 4. Write an inequality to represent the amount Diego can spend on a single pair of socks. Student Response Teachers with a valid work email address can click here to register or sign in for free access to Student Response. Activity Synthesis Here is what we want students to understand as a result of this activity: In order to find the solution to an inequality like $$3n + 25 \geq 68$$, we can solve an equation to find the point where $$3n + 25 = 68$$. This is the point that separates numbers that are solutions to the inequality from numbers that are not solutions. To find whether the solution to the inequality is $$n \geq 14\frac13$$ or $$n \leq 14\frac13$$, we can substitute some values of $$n$$ that are greater than $$14\frac13$$ and some that are less than $$14\frac13$$ to check. Alternatively, we can think about the context: If Andre wants to make more money, he needs to sell more magazines, not fewer. If Diego wants to spend less than$35, he needs to spend less for socks, not more. Ask students:

• How does solving the equation help us solve an inequality? What does the solution tell us about solutions to the inequality?
• What are some ways we can determine whether the solution to an inequality should use less than or greater than?
• How can we check whether a value is a solution to the inequality?
• Could Andre sell exactly $$14\frac13$$ subscriptions?
• Can Diego pay exactly $3.01 for each pair of socks? • How can we tell if there are restrictions on the solutions of the inequality, such as only positive numbers or only whole numbers? 4.3: Granola Bars and Savings (15 minutes) Activity The purpose of this activity is for students to interact with contexts in which the direction of inequality is the opposite of what they might expect if they try to solve like they would with an equation. For example, in the second problem, the original inequality is $$9(7-x) \leq 36$$, but the solution to the inequality is $$x \geq 3$$. Some students might solve the associated equation and then test values of $$x$$ to determine the direction of inequality. That method will be introduced in more generality in the next lesson. This activity emphasizes thinking about the context in deciding the direction of inequality. Launch Keep students in the same groups. Give 5–10 minutes of quiet work time and partner discussion followed by a whole-class discussion. Student Facing 1. Kiran has$100 saved in a bank account. (The account doesn’t earn interest.) He asked Clare to help him figure out how much he could take out each month if he needs to have at least $25 in the account a year from now. 1. Clare wrote the inequality $$\text-12x + 100 \geq 25$$, where $$x$$ represents the amount Kiran takes out each month. What does $$\text-12x$$ represent? 2. Find some values of $$x$$ that would work for Kiran. 3. We could express all the values that would work using either $$x \leq \text{__ or } x \geq \text{__}$$. Which one should we use? 4. Write the answer to Kiran’s question using mathematical notation. 2. A teacher wants to buy 9 boxes of granola bars for a school trip. Each box usually costs$7, but many grocery stores are having a sale on granola bars this week. Different stores are selling boxes of granola bars at different discounts.

1. If $$x$$ represents the dollar amount of the discount, then the amount the teacher will pay can be expressed as $$9(7-x)$$. In this expression, what does the quantity $$7-x$$ represent?
2. The teacher has $36 to spend on the granola bars. The equation $$9(7-x)=36$$ represents a situation where she spends all$36. Solve this equation.
3. What does the solution mean in this situation?