~ 6.14 ~

Finding Solutions to Inequalities in Context

<< Previous

Unit 6

Next >>

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.

Notes

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.

COPY CHART INTO YOUR NOTES

We can use a number line to represent or show inequalities. We use a circle to show the lowest/highest value and then an arrow to show the other numbers that could satisfy that inequality.

For example, to show the inequality x > 2, we would draw a circle at 2, then an arrow going to the right, as x could be any number greater than 2.

To show "greater than", we use an open circle because the value of x is not included. To show "greater than or equal to", we use a filled-in circle because the value of x is included.

If x is between two values, we represent each of these values with the appropriate circle, then join the two circles with a line.

Activities

14.2 Earning Money for Soccer Stuff

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.
  - 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.
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.
  - 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.

14.3 Granola Bars and Savings

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 -12x + 100 ≥ 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 ≤ __ or x ≥ __. Which one should we use?
  - Write the answer to Kiran’s question using mathematical notation.
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?
  - 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 ≥ 3 or x ≤ 3. Which do you think it is? Explain your reasoning.

Summary

No summary video.

Assignment

Check Google Classroom!

Report abuse