LINEAR EQUATIONS AND INEQUALITIES IN ONE VARIABLE

2.8 Solving Linear Inequalities

Key Terms
Use the vocabulary terms listed below to complete each statement in exercises 1-7.

interval interval notation negative infinity

linear inequality in one variable addition property of inequality

multiplication property of inequality three part inequality

1. A(n) ___________________can be written in the form Ax + B < C , Ax + B ≤ C ,
Ax + B > C , or Ax + B ≥ C , where A, B, and C are real numbers , with A ≠ 0.

2. ___________________ is used to indicate all real numbers to the left of a specific
location on a number line.

3. The ___________________ states that the same number can be added to (or subtracted
from ) both sides of an inequality without changing the solution set .

4. ___________________ is a simplified notation that uses parentheses ( ) and/ or brackets
[ ] to describe an interval on a number line.

5. An inequality that says that one number is between two other numbers is called a
___________________.

6. The ___________________ states that both sides of an inequality may be multiplied (or
divided) by a positive number without changing the direction of the inequality symbol .
Multiplying ( or dividing ) by a negative number reverses the inequality symbol.

7. A(n) ___________________ is a portion of a number line.

Objective 1 Graph intervals on a number line.

Graph each inequality on a number line.

Objective 2 Use the addition property of inequality.

Solve each inequality and graph the solutions.

Objective 3 Use the multiplication property of inequality.

Solve each inequality and graph the solutions.

Objective 4 Solve linear inequalities by using both properties in inequality.

Solve each inequality.

Solve each inequality and graph the solutions.

Objective 5 Solve applied problems by using inequalities.

Use an inequality to solve each problem.

21. Faustino sold two antique desks for $280 and $305. How much should he charge for the third in order to average at least $300 per desk?
22. Find every number such that one third the sum of that number and 24 is less than or equal to 10.
23. Lauren has grades of 98 and 86 on her first two chemistry quizzes. What must she score on her third quiz to have an average of at least 91 on the three quizzes?
24. Nina has a budget of $230 for gifts for this year. So far she has bought gifts costing $47.52, $38.98, and $26.98. If she has three more gifts to buy, find the average amount she can spend on each gift and still stay within her budget.
25. Ruth tutors mathematics in the evenings in an office for which she pays $600 per month rent. If rent is her only expense and she charges each student $40 per month, how many students must she teach to make a profit of at least $1600 per month?

Objective 6 Solve linear inequalities with three parts.

Solve each inequality and graph the solutions.