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LESSON 6.4 Solving compound inequalities
Vocabulary:
A compound inequality. Consists of two separate inequalities joined by and or or.
EXAMPLE 1
Write and graph a compound inequality_______________________________
Translate the verbal phrases into an inequality. Then graph the inequality.
a. All real numbers that are less than or equal to 7 or greater than or equal to 10.
Inequality: x ≤ 7 or x ≥ 10
Interval Notation:  ,7 10,  
b. All real numbers that are greater than –1 and less than or equal to 1.
Inequality: –1 < x < 1
Interval Notation:  1,1
Exercises for Example 1
Translate the verbal phrases into an inequality, write it in Interval Notation and then graph the
inequality.
Phrase
All real numbers that are
less than –3 or greater than
0.
All real numbers that are
less than 9 and greater than
or equal to 7.
All real numbers that are
greater than or equal to 14
or less than or equal to 10.
Inequality
Interval Notation
Graph
EXAMPLE 2
Solve a compound inequality with ‘AND’
Solve 7 ≤ x – 4 ≤ 12. Graph your solution.
Solution
7 ≤ x – 4 ≤ 12
Write original inequality.
7 ≤ x – 4 ≤ 12
+4 +4 +4
Add 4 to each expression.
11 ≤ x ≤ 16
Simplify.
11,16
EXAMPLE 3
Solve a compound inequality with ‘OR’
Solve 3x + 4 < 16 or 5x – 12 > 13. Graph your solution.
Solution
Solve the two inequalities separately.
3x + 4 < 16
or
5x – 12 > 13
3x + 4 – 4 < 16 – 4
or
5x – 12 + 12 > 13 + 12
3x < 12
or
3 x 12

3
3
or
5x > 25
5 x 25
x5  5
x<4
or
x>5
Write original inequality
Use addition or subtraction
property of inequality
Simplify
Use division property of
inequality
Simplify.
The solutions are all real numbers less than 4 or greater than 5.
 , 4 5, 
Exercises for Examples 2 and 3_______________________________________
Solve the inequality. Write the Answer in Interval Notation. Graph your solution.
1.
9 < 2x + 3 < 15
2. 30 ≥ –7x – 12 > 16
3.
28 ≤ 4(2x – 3) ≤ 68
4. 3x – 7 < 11 or –x + 4 < -15
5.
x + 18> 6 or 7x + 5 < –51
6. 3x + 8 > 7x – 12 or 9(x – 2) > 8x – 9
6.4 Solving Compound Inequalities: Class Examples
Solve the Inequality. Write your answer in Interval Notation. Graph your solution.
1.
2.
3.
4.
5.
6.
7.
8.
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6.4 Homework
Translate the verbal phrase into an inequality. Then graph the inequality. Write the inequality in
interval notation.
1. All real numbers that are less than or equal to -8 or greater than 12.
Inequality:___________________
Interval Notation:_____________________
1
2. All real numbers that are greater than or equal to 7 and less than or equal to 10.
2
Inequality:___________________
Interval Notation:________________________
Write and graph an inequality that describes the situation. Write the inequality in set notation and
interval notation.
3. The temperature inside a room is uncomfortable if the temperature is lower than 60 F or higher than 75 F.
Inequality:___________________
Interval Notation:_____________________
4. The minimum speed on the highway is 40 miles per hour, and the maximum speed is 60 miles per hour.
Inequality:___________________
Interval Notation:_____________________
Solve the inequality. Graph your solution and write it in set and interval notation.
5. 7  y  8  12
Interval Notation:_____________________
6. 6  3n  9  21
Interval Notation:_____________________
2
7. 7  (6q  18)  9
3
Interval Notation:_____________________
8. 16  s  6 or
9.
2s  5  11
14  w  3 or 5w 13  w  7
10. 2h  7  h  5 or
1
(h  8)  9
4
Interval Notation:_____________________
Interval Notation:_____________________
Interval Notation:_____________________