1.4 Solving with Absolute Value
Absolute Values won't
put up with negativity.
Absolute Value
The distance a value is from zero
|-5| = 5
|5| = 5
Put any value but zero behind
bars and the results will always
be positive!
|3| = 3
|-7| = 7
7 units
5 units
5 units
3 units
-10
-9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
When we find distance we subtract...
The distance between 9 and 7 is 2
9-7=2
7 units
2 units
-10
-9
-8
-7
-6
-5 -4
-3
-2
-1
The distance between 3 and -4 is 7
3 - (-4) = 7
0
1
2
3
4
5
6
7
8
9
10
Example 1
|f|=2
or | f - 0 | = 2
Means the distance
-10
-9
-8
-7
-6
Case 2
f = -2
between f and 0 is 2.
Case 1
f=2
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
10
9
| f + 5| = 2
Example 2
or
| f - (-5)| = 2
Means the distance
between f and -5 is 2.
Case 1
Case 2
f+5=2
-5 -5
f = -3
-10
-9
-8
-7
-6
f + 5 = -2
-5 -5
Subtract 5 from
each side.
f = -7
-5 -4
-2
-3
-1
0
1
2
3
4
5
6
7
8
9
10
Example 3
| f + 5| = -8
Means the distance between
f and -5 is -8. Since distance
cannot be negative, the solution
is the empty set ∅ (or no solution).
-10
-9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
If | x | = 4
So, if
then x = 4 or x = -4
then
or
| 2x - 3| - 2 = 7
Solving Steps for Absolute Value Equations
+2 +2
1. Isolate the absolute value.
| 2x - 3| = 9
2. Create 2 equations
a. The expression inside the absolute
value bars can be a positive value 2x - 3 = 9 or 2x - 3 = -9
or a negative (or opposite) value.
2x = 12
2x = -6
3. Solve each equation. The x values are
x=6
x = -3
one OR the other.
4. Check for extraneous solutions
| 2(6) - 3| - 2 = 7
(solutions that do not satisfy the
equation). They are not included in the
| 2(-3) - 3| - 2 = 7
final solution.
You try 1
You try 2
You try 3
You try 4
You try 4
You try 5
2x + 1 = x + 5
2x + 1 = -(x + 5)
You try 6
Absolute Value Expressions
Expressions with absolute values defines an upper and lower range in which
a value must lie.
Write an Absolute Value Equation
Example 4
Ice cream should be stored at 10oF with an allowance for 4o. Write and solve
an equation to find the maximum and minimum temperatures at which the ice
cream should be stored.
4o
4o
6o
x - 10 = -4
x =6
10
o
| x - 10| = 4
14o
x - 10 = 4
x = 14
Write an Absolute Value Equation
In the month of January, we are predicted to get 43 inches of snow, give or
take 8 inches. Write an absolute value equation to represent the snowfall in
January.