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Math 1314
Section1.7 Notes
Absolute Value Equations and Inequalities
Interval Notations
Absolute Value Equations
Recall:
Definition: The absolute value of x denoted |x| and read “Absolute value of x” is defined as
follow:
if x  0
x
x 
if x  0
 x
Remark: Absolute value of x is also considered the distance from zero to x on the real number
line.
Examples:
 If x = 7 then | x | = | 7 | = 7.
 If x = - 7 then | x | = | -7 | = 7.
 The opposite numbers have the same absolute value i.e. | x | = | -x |.
 If | y | = 7 then y = 7 or y = - 7.
 Absolute value of a real number is nonnegative or | x | ≥ 0 where x is a real number.
Absolute Value Equations. | A | = B and | A | = | B | where A and B are algebraic expressions.
Formulas:
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A  B and B  0 then A  B or A  - B
A  B and B  0  No solutions
A  B  A  B or A  - B
Steps for Solving Absolute Value Equations | A | = B.
1. Isolate the absolute value expression.
2. There are two special cases:
- If B < 0 --- NO solution.
- If B ≥ 0, remove the absolute value sign by writing as 2 equations A = B or A = - B.
3. Solve for the variable.
Examples: Sole the following equations.
1. | x – 4 | = 9
2. | 4x + 1 | + 5 = 6
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3. | 3x + 8 | = – 3
4. 3| 3x + 8 | + 2 = 8
5. 4| x – 2 | + 7 = 3
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6. 15 – |2x – 3| = 7
7. | 2x + 5 | = | x + 4|
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Absolute Value Inequalities:
Formulas: For B is positive.
A  B  A  B or
A  B
A  B  B  A  B
A   B  all real numbers or S   ,  
A   B  no solution or S  
For B is zero
A  0  all real numbers or
S   ,  
A  0  A  0 or all real numbers except A  0
A 0 A0
A  0  no solution or S  
Examples: Solve the following inequalities. Graph the solutions on the number line and write the
solutions set in interval notation.
1. x  2
-2
0
2
Closer: that means the distance from zero to x is either LESS THANor EQUAL
to 2 units.
-2 ≤ x ≤ 2 (Compound inequality)
Thus S = [-2, 2]
2.
x 5
Further away: that means the distance from zero to x is GREATER THAN 5 units.
x < -5(to the left)
x > 5 (to the right)
Thus S = (-∞, -5) U ( 5, ∞) .
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3.
x 8
4.
x  11
5.
3 x  12
6.
2x  1  7
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7.
3x  2  3  9
8. 3 9  5 x  5  8
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9. 2 3x  5  5  7
10. 3 x  2  0
11. 3 x  7  0
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12. 5 x  1  4  4
13. 5  3 4 x  1  5
14. x  5  7
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