Lesson 14

Finding Solutions to Inequalities in Context

14.1: 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.

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 \(2x > \text-20\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

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.

14.2: 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 \(\geq\).

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

Student responses to this activity are available at one of our IM Certified Partners

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?

14.3: 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?
    4. The teacher does not have to spend all \$36. Write an inequality relating 36 and \(9(7-x)\) representing this situation.
    5. The solution to this inequality must either look like \(x \geq 3\ \text{or } x \leq 3\). Which do you think it is? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Jada and Diego baked a large batch of cookies.

  • They selected \(\frac14\) of the cookies to give to their teachers.
  • Next, they threw away one burnt cookie.
  • They delivered \(\frac25\) of the remaining cookies to a local nursing home.
  • Next, they gave 3 cookies to some neighborhood kids.
  • They wrapped up \(\frac23\) of the remaining cookies to save for their friends.

After all this, they had 15 cookies left. How many cookies did they bake?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of the discussion is to let students voice their reasoning about the direction of the inequality by reasoning about the context. Ask students to share their reasons for choosing the direction of inequality in their solutions. Some students may notice that the algebra in both problems involves multiplying or dividing by a negative number. Honor this observation, but again, the goal is not to turn this observation into a rule for students to memorize and follow. Interpreting the meaning of the solution in the context should remain at the forefront.

As students model real-world situations, questions about the interpretation of the mathematical solution should continue to come up in the conversation. For instance, the amount of the granola bar discount cannot be \$3.5923, even though this is a solution to the inequality \(x \geq 3\). The value -10 is a solution to Kiran’s inequality, even though he can’t withdraw -10 dollars. Students can argue that negative values for \(x\) simply don’t make sense in this context. Some may argue that we should interpret \(x=\text-10\) to mean that Kiran deposits \$10 every month.

Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention
Representing, Writing, Conversing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After a student shares their reasons for the direction of the inequality in their solution, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. This will help students understand how context determines the direction of inequality.
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

By the time students have finished this lesson, they have reasoned about solutions to several inequalities, all of which involve some kind of final decision about the direction of inequality. Return to the ideas in the warm-up for the previous lesson. Draw the number line showing solutions to \(x > 1\) on the board. Ask students to name some values of \(x\) that satisfy the inequality. For each of those values of \(x\), plot the value of \(\text-x\) on the number line together (perhaps in a different color). What inequality did we just graph?

14.4: Cool-down - Colder and colder (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Suppose Elena has \$5 and sells pens for \$1.50 each. Her goal is to save \$20. We could solve the equation \(1.5x + 5 = 20\) to find the number of pens, \(x\), that Elena needs to sell in order to save exactly \$20. Adding -5 to both sides of the equation gives us \(1.5x = 15\), and then dividing both sides by \(1.5\) gives the solution \(x=10\) pens.

What if Elena wants to have some money left over? The inequality \(1.5x + 5 > 20\) tells us that the amount of money Elena makes needs to be greater than \$20. The solution to the previous equation will help us understand what the solutions to the inequality will be. We know that if she sells 10 pens, she will make \$20. Since each pen gives her more money, she needs to sell more than 10 pens to make more than \$20. So the solution to the inequality is \(x > 10\).