# Solve Multi-step Inequalities

You have learned to solve multi-step equations. Now, using the same process by applying inverse operations, you can apply those skills to solving multi-step inequalities. Remember to reverse the inequality symbol when you multiply or divide by a negative number, but everything else is exactly the same for inequalities as equations.

Take a look at this problem:

Mrs. Holland brings \$200 to a fundraiser at the school.  She wants to leave the event with at least \$50 in her purse.  Visitors at the fundraiser buy raffle tickets for several different prizes. Each raffle ticket costs \$6.  How many raffle tickets can Mrs. Holland buy and still leave with at least \$50 in her purse?

1. How much money does Mrs. Holland have at the start of the fundraiser?
2. Let r = the number of raffle tickets purchased.   Write an expression to show how much it costs to buy r tickets.
3. Use the expression above to write a different expression that shows how much money Mrs. Holland would have left after buying r tickets.
4. Suppose Mrs. Holland buys 30 tickets. How much money would she have left?
5. Suppose Mrs. Holland buys 25 tickets. How much money would she have left?

The greatest number of tickets that she can purchase and still have \$50 is 25; however, she can buy fewer tickets and still have at least \$50 left. This problem is an example of a two-step inequality word problem.

Strategy – Translate the words to math.

(Starting Amount) – (Raffle ticket price)(Number of tickets) ≥ (is greater than/equal to) (Amount left)

Solution Set means she can purchase 25 or fewer tickets and still have \$50 left.

Investigate

Julie is building a game room in her basement.  She wants the width of the room to be 24 feet and the length to be longer than the width.  If she wants the area of the room to be more than 700 square feet, what could be the length?  Use the diagram Julie drew to help you write and solve an inequality to solve this problem.

Strategy

Restate the problem so you can translate key phrases to an inequality.

The product (area) of the width and length must be greater than 700 square feet.

The length of Julie’s game room must be at least 29.17 feet.

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