Lesson 4Finding Solutions to Inequalities in Context

Learning Goal

Let’s solve more complicated inequalities.

Learning Targets

I can describe the solutions to a inequality by solving a related equation and then reasoning about values that make the inequality true.
I can write an inequality to represent a situation.

Lesson Terms

solution to an inequality

Warm Up: Solutions to Equations and Solutions to Inequalities

Solve $- x = 10$
Find 2 solutions to $- x > 10$
Solve $2 x = - 20$
Find 2 solutions to $2 x > - 20$

Activity 1: Earning Money for Soccer Stuff

Problem 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.

Let $n$ represent the number of magazine subscriptions Andre sells this week. Write an expression for the amount of money he makes this week.
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.
If Andre sold 16 magazine subscriptions this week, would he reach his goal? Explain your reasoning.
What are some other numbers of magazine subscriptions Andre could have sold and still reached his goal?
Write an inequality expressing that Andre wants to make at least $68.
Write an inequality to describe the number of subscriptions Andre must sell to reach his goal.

Problem 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.

Let $x$ represent the price of one pair of socks. Write an expression for the total cost of the socks and shorts.
Write and solve an equation that says that Diego spent exactly $35 on the socks and shorts.
List some other possible prices for the socks that would still allow Diego to stay within his budget.
Write an inequality to represent the amount Diego can spend on a single pair of socks.

Activity 2: Granola Bars and Savings

Problem 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.

Clare wrote the inequality $- 12 x + 100 \geq 25$ , where $x$ represents the amount Kiran takes out each month. What does $- 12 x$ represent?
Find some values of $x$ that would work for Kiran.
We could express all the values that would work using either
$x \leq$ or $x \geq$ .
Which one should we use?
Write the answer to Kiran’s question using mathematical notation.

Problem 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.

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

Are you ready for more?

Jada and Diego baked a large batch of cookies.

They selected $\frac{1}{4}$ of the cookies to give to their teachers.
Next, they threw away one burnt cookie.
They delivered $\frac{2}{5}$ of the remaining cookies to a local nursing home.
Next, they gave 3 cookies to some neighborhood kids.
They wrapped up $\frac{2}{3}$ of the remaining cookies to save for their friends.

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

Lesson Summary

We use inequalities to describe a range of numbers. In many places, you are allowed to get a driver’s license when you are at least 16 years old. When checking if someone is old enough to get a license, we want to know if their age is at least 16. If $h$ is the age of a person, then we can check if they are allowed to get a driver’s license by checking if their age makes the inequality $h > 16$ (they are older than 16) or the equation $h = 16$ (they are 16) true. The symbol $\geq$ , pronounced “greater than or equal to,” combines these two cases and we can just check if $h \geq 16$ (their age is greater than or equal to 16). The inequality $h \geq 16$ can be represented on a number line:

A number line from 0 to 28 with a closed circle at 16 an an arrow extending to the right to 28.

To compare $=$ , $>$ , and $\geq$ , let’s consider a situation three ways. Suppose Elena has $5 and sells pens for $1.50 each. Her goal is to save $20. We could solve the equation $1.5 x + 5 = 20$ to find the number of pens, $x$ , that Elena needs to sell in order to save exactly $20. Adding -5 to each side of the equation gives us $1.5 x = 15$ , and then dividing each side by 1.5 gives the solution $x = 10$ pens.

What if Elena wanted to save more than $20? The inequality $1.5 x + 5 > 20$ 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 $x > 10$ .

What if Elena wanted to save at least $20? The inequality $1.5 x + 5 \geq 20$ tells us that the amount of money Elena makes needs to be at least $20. The solution to this inequality is $x \geq 10$ .