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Chapter 2.7 – Absolute Value Inequalities
Objectives
Solve absolute value inequalities of the form /x/ < a
Solve absolute value inequalities of the form /x/ > a
Example 1
Solve: ￨ x ￨ 3
[ ]
The solution set is {-3, 3}
-5 0-1-2-4 -3 4 3 2 1 5
Solving Absolute Value expression of the form ￨ x ￨ < a If a is a positive number, then ￨ x ￨ < a
is equivalent to –a < x< a
Example 2:
Solve for m: ￨m – 6 ￨ < 2
Step 1: -a < x < a
Replace x with m - 6 and a with 2
-2 < m – 6 < 2
Example 2 continued
Solve the compound inequality
-2 < m – 6 < 2
-2 + 6 < m – 6 + 6 < 2 + 6
4 < m < 8
The solution set is (4, 8) and its graph is,
Give it a try!
Solve ￨ x-2 ￨ 1
HINT
Before using an absolute value inequality property, you MUST MUST ISOLATE the absolute value ISOLATE the absolute value expressionexpression on one side of the inequality!
Example 3:
Solve for x: ￨ 5x + 1 ￨ + 1 10
Give it a try!
Solve: ￨ 2x - 5 ￨ + 2 9
Example 4
Solve for x:
The absolute value of a number is always nonnegative and can never be less than –13. This inequality has NO solution! The solution set is { } or 0
1310
12 x
Give it a try!
Solve: 47
25 x
Let ￨ x ￨ 3
The solution set includes all numbers who distance from 0 is 3 or more units. The graph of the solution set contains 3 and all points to the right of 3 on the number line or –3 and all points to the left of –3 on the number line..
The solution is ( - , -3] U [3, )
Form of ￨ x ￨ > a
Solving Absolute Value Inequalities of the form ￨ x ￨ > a If a is a positive number, the ￨ x ￨ > a
is equivalent to x < -a OR x > a
Example 5
Solve for y:
Step 1: Rewrite the inequalities without absolute value bars
y – 3 < -7 or y – 3 > 7Step 2: Solve the compound inequality
y < -4 or y > 10
3 7y
Example 5 Continued
Step 3: Graph the inequalities
Step 4: Write the solution set
(-∞, -4) U (10, ∞)
Give it a try!
Solve: 2 4x
Example 6: Isolate the Absolute Value expression!
Solve:
Step 1: Isolate the Absolute Value Expression
2 9 5 3x
52 59 5 3x
2 9 2x
Example 6 - Continued
***Remember***
The absolute value of any number is always a nonnegative and thus is always great than -2. The inequality and the original inequality are true for all values of x. The solution set is {x/ x is a real number} or (-∞,∞)
Example 7: Isolate the Absolute Value Expression!
Solve:
Step 1: Isolate the Absolute Value Expression
1 7 53x
73
71 7 5x
13
2x
Example 7 - Continued
Step 2: Write the absolute value inequality as an equivalent compound inequality.
1 2 1 23 3
1 33 3
3 9
x xor
x xor
x or x
Example 7 continued
Step 3: Solve each inequality
1 2 1 23 3
1 33 3
3 9
x xor
x xor
x or x
Example 7 continued
Step 4: Write the solution set and graph
(-∞, -3] U [9, ∞)
Give it a try!
Solve: 3 8 45x
Example 8 - Zero
Solve for x:
**Remember – the absolute value of any expression will never be less than 0, but it may be equal to 0. Thus solve the equation by setting it equalequal to zero!
2 10
3
x
Example 8 Continued 2 1
03
x
2( 1)0
3
x
2( 1)0
33 3
x
2 2 0x 2 2x
1x
You give try!
Solve: 5( 3)
02
x
Chart on Page 113
Copy the chart from 113 into your notes!