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 eac 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 � x - 1 � 0
5. 3 < 15 + x < 10 6. -12 < x - 3 � 1