# Solving Inequalities

Video solutions to help Grade 7 students learn how to solve word problems leading to inequalities.

## New York State Common Core Math Module 3, Grade 7, Lesson 14

### Lesson 14 Student Outcomes

• Students solve word problems leading to inequalities that compare px + q and r, where p, q, and r are specific rational numbers.
• Students interpret the solutions in the context of the problem.

Lesson 14 Summary

The goal to solving inequalities is to use If-then moves to make 0s and 1s to get the inequality into the form x > a number or x < a number. Adding or subtracting opposites will make 0s. According to the If-then move, a number that is added or subtracted to each side of an inequality does not change the solution of the inequality. Multiplying and dividing numbers makes 1s. A positive number that is multiplied or divided to each side of an inequality does not change the solution of the inequality. However, multiplying or dividing each side of an inequality by a negative number does reverse the inequality sign.

Given inequalities containing decimals, equivalent inequalities can be created which have only integer coefficients and constant terms by repeatedly multiplying every term by ten until all coefficients and constant terms are integers.

Given inequalities containing fractions, equivalent inequalities can be created which have only integer coefficients and constant terms by multiplying every term by the least common multiple of the values in the denominators.

Lesson 14 Classwork

Opening Exercise
You are the owner of the biggest and newest rollercoaster called the 'Gentle Giant'. The rollercoaster costs \$6 to ride. The operator of the ride must \$200 pay per day for the ride rental and \$65 per day for a safety inspection. If you want to make a profit of \$1000 at least each day, what is the minimum number of people that must ride the rollercoaster to make that profit? Write an inequality that can be used to find the minimum number of people, p, that must ride the rollercoaster each day to make the daily profit.
Solve the inequality.
Interpret the solution.

Example 1
A youth summer camp has budgeted \$2000 for the campers to attend the carnival. The cost for each camper is \$17.95, which includes general admission to the carnival and meals. The youth summer camp must also pay \$250 for the chaperones to attend the carnival and \$350 for transportation to and from the carnival. What is the greatest amount of campers that can attend the carnival if the camp must stay within their budgeted amount?

Example 2
The carnival owner pays the owner of an exotic animal exhibit \$650 for the entire time the exhibit is displayed. The owner of the exhibit has no other expenses except for a daily insurance cost. If the owner of the animal exhibit wants to make more than \$500 in profits for the 5 1/2 days, what is the greatest daily insurance rate he can afford to pay?

Example 3
There are several vendors at the carnival who sell products and also advertise their businesses. Shane works for a recreational company that sells ATVs, dirt bikes, snowmobiles, and motorcycles. His boss paid him \$500 for working all of the days at the carnival plus 5% commission on all of the sales made at the carnival. What was the minimum amount of sales Shane needed to sell if he earned more than \$1,500?