Download 6-5 Solving Absolute Value Equations

6-5 Solving Absolute Value
Equations
To learn how to solve Absolute Value Equations
What is Absolute Value?
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The Absolute Value of a number is its
distance from zero on a number line.
The symbol 𝑎 represents the absolute
value of a.
Remember, since you are measuring
distance the absolute value of a number will
always be positive.
Absolute Value Equations
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Consider the case 𝑥 = 5.
This means the Distance between 0 and x is 5 units.
5 is five units from 0
-5 is also five units from 0
So 𝑥 = 5 𝑜𝑟 𝑥 = −5
There are two solutions!
Absolute Value of a Number
So there are two cases…
Case 1:
 The value inside the
absolute value symbol can
be positive.
Case 2:
 The value inside the
absolute value symbol can
be negative.
Ex 2: Solve an Absolute Value Equation
Solve x -18 = 5
Abs.Value equals a Positive
Abs.Value equals a Negative
x -18 = 5
x -18 +18 = 5 +18
x = 23
Check
x -18 = -5
x -18 +18 = -5 +18
x = 13
Check
23 -18 = 5
13-18 = 5
5 =5
-5 = 5
5=5
5=5
So x = 23 or x =13
Ex 2: Solve an Absolute Value Equation
Solve y + 3 = 8
Abs.Value equals a Positive
Abs.Value equals a Negative
y+3=8
y + 3- 3 = 8- 3
x=5
Check
y + 3 = -8
y + 3 - 3 = -8 - 3
x = -11
Check
5+3 = 8
-11+ 3 = 8
8 =8
-8 = 8
8=8
8=8
So x = -11 or x = 5
Ex 2B: Solve an Absolute Value Equation
Solve 5x - 6 - 9 = 0
5x - 6 - 9 + 9 = 0 + 9
5x - 6 = 9
5x - 6 = 9
5x - 6 + 6 = 9 + 6
5x = 15
x=3
5x - 6 = -9
5x - 6 + 6 = -9 + 6
5x = -3
-3
x=
5
Solving Absolute Value Equations
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Isolate Absolute Value on one side of the equation.
Set up two equations one equals a positive distance one
equals a negative distance.
Solve for the variables.
Check your answers.
Ex 3: No Solution
Solve 2x - 4 + 9 = 0
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2x - 4 + 9 - 9 = 0 - 9
2x - 4 = -9
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Because Absolut Value
measures distance, we can
never take an absolute
value and get a negative
value.
Because my absolute value
expression equals a
negative integer, there is
no solution to this
problem.
Checking Solutions…
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It is important to check your solutions when solving
absolute value equations.
Sometimes the answers may not be the correct
solution to the original equation.
Ex 4: One Solution
Solve x + 6 = 3x - 2. Check your Solutions.
x + 6 = 3x - 2
6 = 2x - 2
8 = 2x
4=x
x=4
x + 6 = - ( 3x - 2 )
x + 6 = -3x + 2
6 = -4x + 2
4 = -4x
-1 = x
x = -1
So far it looks like my solutions are 4 and -1.
Ex 4: One Solution(Check)
Solve x + 6 = 3x - 2. Check your Solutions.
4 + 6 = 3(4) - 2
-1+ 6 = 3(-1) - 2
4 + 6 = 12 - 2
5 = -3- 2
10 = 12 - 2
5 = -5
10 = 10
5 ¹ -5
-1 is not a solution
4 is a solution
So my solution is 4.