6.4 Solve Compound Inequalities

Goal  Solve and graph compound inequalities.

Your Notes

VOCABULARY
Compound inequality
A compound inequality consists of two separate inequalities joined by and or or.

Example 1
Write and graph compound inequalities

Translate the verbal phrase into an inequality. Then graph the inequality.
a.	All real numbers that are greater than or equal to 2 and less than 2.
b.	All real numbers that are less than or equal to 3 or greater than 6.
c.	All real numbers that are greater than 8 and less than or equal to 3.
Solution
a.	2 __ x __ 2

b.	x __ 3 or x __ 6

c.	8 __ x __ 3

Your Notes

Example 2
Solve a compound inequality with and

Solve 15 ’ 3x  3 < 24. Graph your solution.

Solution
Separate the compound inequality into two inequalities. Then solve each inequality separately.
15 3x  3 and 3x  3  24	Write two inequalities.	
_18_ 3x and 3x  _27_	Add _3_ to each expression.	
_6_ x and x  _9_	Divide each expression by _3_.	
The compound inequality can be written as _6 ' x < 9_
The solutions are all real numbers _greater than or equal to 6_ and _less than 9_.

Example 3
Solve a compound inequality with and

Solve 15 < 7x + 1 < 50. Graph your solution.

Solution
15 < 7x + 1 < 50	Write original inequality.	
_14_ < 7x < _49_	Subtract _1_ from each expression.	
_2_ > x > _7_	Divide each expression by _7_ and _reverse both inequality symbols_.	
The solutions are all real numbers _greater than 7_ and _less than 2_.

Your Notes

Example 4
Solve a compound inequality with or

Solve 5x + 6 ' 9 or 2x   8 > 12. Graph your solution.

Solution
5x + 6 ' 9 or 2x   8 > 12	Write original inequality.	
5x '_15_ or 2x > _20_	Use addition or subtraction property of inequality.	
x '_3_ or x > _10_	Use division property of inequality.	
The solutions are all real numbers _less than or equal to 3_ or _greater than 10_.

Checkpoint Solve the inequality. Graph your solution.

1.	3 ‘ 2x + 1  11
2” x  5

2.	9x +1  17 or 7x  12  9
x  2 or x  3

Homework
_____________________________________________________________________________________

_______________________________________________________________________