1. 1.6 Absolute Value Equations and Inequalities

2. Absolute Value must be by itself on the left side.
Absolute value: is a number’s distance from zero on the number line and it is nonnegative
If x  0, then  x  = x or If x  0, then  x  = -x
Distance cannot be negative!!!
8 = 8
– 8 = 8 8 8
Note:
Absolute Value must be by itself on the left side.

3. Solving Absolute Value Equations
This has two solutions, since it can be equal to – 6 or 6,
because  – 6 = 6 and 6 = 6.
1. Solve 15 – 3x = 6.
15 – 3x = 6 or
– –15
– 3x = – 9
– – 3
x = 3
Check: 15 – 3x = 6
15 – 3(3) = 15 – 3(7) = 6
6 = –6 = 6
15 – 3x = – 6
–15
–15
–3x =
–21
– 3
– 3 or
x = 7
{3, 7}

4. Extraneous Solutions -5
Extraneous Solutions: a solution of an equation derived from an original equation that is not a solution of the original equation
2. Solve 3x – 4 = -4x – 1
3x – 4 = -4x – 1 or 3x – 4 = –(-4x – 1)
7x = 3
x = 3/7
3(3/7) – 4 = -4(3/7) – 1 or 3(-5) – 4 = -4(-5) – 1
-15 – 4 = 20 – 1
19 = 19; true, solution
or 3x – 4 = 4x + 1
or –5 = x √:
(9/7) – 4 = (-12/7) – 1
19/7 = –19/7;
false, extraneous solution -5

5. Absolute Value must be by itself on the left side.
Absolute Value Inequalities
Let k represent a positive real number.
x > k is equivalent to x < -k or x > k.
x  k is equivalent to -k  x  k.
Outsides
Insides
Note:
Absolute Value must be by itself on the left side.

6. Solving Inequalities of the Form A  b
3. Solve -2x + 1 + 5 ≥ -3. Graph the solution x + 1 ≥ x + 1  4 -4  x + 1   x  3
Flip sign when you divide by a negative
Graph

7. Solving Inequalities of the Form A > b
4. Solve 2x - 5  3. Graph the solution.
2x – 5 < -3 or 2x – 5  3
+ 5 < >+ 5
2x < 2 2x > 8
x < or x > 4
Graph

8. True All Real numbers Distance cannot be negative!!!
Translation: “The distance from zero on the number line is greater than -3”.
Distance cannot be negative!!!
The smallest distance would be zero.
True
All Real numbers

9. False No Solution No Graph Distance cannot be negative!!!
Translation: “The distance from zero on the number line is less than -3”.
Distance cannot be negative!!!
The smallest distance would be zero.
False
No Solution
No Graph

10. Solve and graph.
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