Solving Inequalitites Using Addition and Subtraction

Solving Inequalitites Using Addition and Subtraction

English Phrase	Mathematical Phrase
at most x	≤ x
at least x	≥ x
5 less than x	x - 5
5 more than x	x + 5
5 is less than x	5 < x
x is between 4 and 6	4 < x < 6 or x > 4 and x < 6

1. Write a mathematical phrase for each of the following English phrases.

a. 8 is no more than x.

b. x is no less than 3.

c. x is not between 6 and 12.

d. x cannot exceed -5.

2. For what values of x, if any, is the mathematical phrase 5 > x > 7 true?

To solve compound inequalities, you must perform the same operation on each of the three parts of the inequality.

Example: Solve the compound inequality 3 < x - 2 < 5.

3 < x - 2 < 5

3 + 2 < x - 2 + 2 < 5 + 2

To get x by itself, add 2 to each part.

5 > x > 7

Thus, x is all numbers between 5 and 7.

Solve each compound inequality.

3.
-2 < x + 1 < 4
4.
-4 ≤ - 1 ≤ 0

5. 3 < 15 + x < 10 6. -12 < x - 3 ≤ 1

Example:	Solve the compound inequality 3 < x - 2 < 5.
	3 < x - 2 < 5
	3 + 2 < x - 2 + 2 < 5 + 2	To get x by itself, add 2 to each part.
	5 > x > 7
	Thus, x is all numbers between 5 and 7.

3.	-2 < x + 1 < 4	4.	-4 ≤ - 1 ≤ 0
5.	3 < 15 + x < 10	6.	-12 < x - 3 ≤ 1