6.3.2: Finding Solutions to Inequalities in Context
Lesson
Let's solve more complicated inequalities.
Exercise \(\PageIndex{1}\): Solutions to Equations and Solutions to Inequalities
- Solve \(-x=10\)
- Find 2 solutions to \(-x>10\)
- Solve \(2x=-20\)
- Find 2 solutions to \(2x>-20\)
Exercise \(\PageIndex{2}\): Earning Money For Soccer Stuff
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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.
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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.
Exercise \(\PageIndex{3}\): Granola Bars and Savings
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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 \(-12x+100\geq 25\), where \(x\) represents the amount Kiran takes out each month. What does \(-12x\) represent?
- Find some values of \(x\) that would work for Kiran.
- We could express all the values that would work using either \(x\leq\underline{\quad}\) or \(x\geq\underline{\quad}\). Which one should we use?
- Write the answer to Kiran’s question using mathematical notation.
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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?
Summary
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 \(x>10\).
Glossary Entries
Definition: Solution to an Inequality
A solution to an inequality is a number that can be used in place of the variable to make the inequality true.
For example, 5 is a solution to the inequality \(c<10\), because it is true that \(5<10\). Some other solutions to this inequality are 9.9, 0, and -4.
Practice
Exercise \(\PageIndex{4}\)
The solution to \(5-3x>35\) is either \(x>-10\) or \(-10>x\). Which solution is correct? Explain how you know.
Exercise \(\PageIndex{5}\)
The school band director determined from past experience that if they charge \(t\) dollars for a ticket to the concert, they can expect attendance of \(1000-50t\). The director used this model to figure out that the ticket price needs to be $8 or greater in order for at least 600 to attend. Do you agree with this claim? Why or why not?
Exercise \(\PageIndex{6}\)
Which inequality is true when the value of \(x\) is -3?
- \(-x-6<-3.5\)
- \(-x-6>3.5\)
- \(-x-6>-3.5\)
- \(x-6>-3.5\)
(From Unit 6.3.1)
Exercise \(\PageIndex{7}\)
Draw the solution set for each of the following inequalities.
1. \(x\leq 5\)
2. \(x<\frac{5}{2}\)
(From Unit 6.3.1)
Exercise \(\PageIndex{8}\)
Write three different equations that match the tape diagram.
(From Unit 6.1.3)
Exercise \(\PageIndex{9}\)
A baker wants to reduce the amount of sugar in his cake recipes. He decides to reduce the amount used in 1 cake by \(\frac{1}{2}\) cup. He then uses \(4\frac{1}{2}\) cups of sugar to bake 6 cakes.
- Describe how the tape diagram represents the story.
- How much sugar was originally in each cake recipe?
(From Unit 6.1.2)
Exercise \(\PageIndex{10}\)
One year ago, Clare was 4 feet 6 inches tall. Now Clare is 4 feet 10 inches tall. By what percentage did Clare’s height increase in the last year?
(From Unit 4.3.3)