Solve Multistep Inequalities

image of 4 inequality symbols 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?

Questions to Ask

How much money does Mrs. Holland have at the start of the fundraiser?

Solution: $200.00

Let r = the number of raffle tickets purchased. Write an expression to show how much it costs to buy r tickets.

Solution: 6r

Use the expression above to write a different expression that shows how much money Mrs. Holland would have left after buying r tickets.

Solution: 200-6r

Suppose Mrs. Holland buys 30 tickets. How much money would she have left?

Solution: $20.00

Suppose Mrs. Holland buys 25 tickets. How much money would she have left?

Solution: $50.00

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)

$LaTeX: 200-6r\ge50 \\ \text{Now solve using inverse operations.}\\ \text{Subtract 200 from both sides.}\\ 200-6r\ge50 \\ -200\:\:\:-200\\ \text{Divide both sides by -6 and reverse the symbol.}\\ \frac{-6r}{-6}\ge \frac{-150}{-6}\\ r\le 25$

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

Before we go deeper, watch the video as it shows you how to construct multi-step inequalities from word problems and solve them.

Investigate

image of rectangle with sides x+24 and 24

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.

$LaTeX: 24\left(24+x\right)>700\\ \text{Distribute the 24}\\ 576+24x> 700\\ \text{Subtract 576 from both sides}\\ 24x> 124\\ \text{Divide both sides by 24(You do NOT need to reverse the symbol.)}\\ x> 5\frac{1}{6}\:\:\text{Interpret the solution.}$

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

In the following video, you can see how to solve multistep inequalities that involve combining like terms and performing subtraction and addition. Watch carefully as this demonstrates the same steps as for solving multistep equations.

Solve Multistep Inequalities Practice

Which of the following ordered pairs, (x, y), are solutions of y < 4x - 3? (-1, 1), (2, 5) (-(1/2), -7) (0, -4) text annotation indicator

- Solution: (-(1/2), -7) (0, -4)

Which of the following ordered pairs are solutions of y ≥ 6 - 2x? (0, 0) (1, 5) (3, 6) ((1/2), 8) text annotation indicator

- Solution: (1, 5) (3, 6) ((1/2), 8)

Which of the following ordered pairs are solutions of 4x + 2y ≤ 20? (3, 0) (0, 12) (5, 5) (2, -4) text annotation indicator

- Solution: (3, 0) (2, -4)

Note: In 6^th grade, you learned to graph inequalities and to identify graphs from inequality solution sets. See the image below as it will demonstrate an example of two types of inequalities. As you solve inequalities, remember that you can write the inequality and you may also show the solution set by graphing it on a number line.

Graphing Inequalities examples. A closed dot means the inequality has an "equal to", and an open dot means the inequality is "not equal to".

If you need a deeper review, watch this video to give you a few more examples.

Solve Multistep Inequalities Practice

Now that you have spent some time learning strategies for solving multistep inequalities, you are ready to complete your Inequalities: Solve Multistep Inequalities Practice. Download your practice by CLICKING HERE.

Once you have completed your practice, AND MAKE SURE YOU ATTEMPTED AND WORKED THE PROBLEMS OUT ON YOUR OWN, click here to download your practice key.

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