Solve Equations with Absolute Value
This is a tutorial on solving equations with absolute value. Detailed solutions and explanations are included.
Example 1: Solve the equation
Solution to Example 1:

If x + 6  = 7, then
a)x + 6 = 7
or
b)x + 6 = 7

Solve equation a)
x + 6 = 7
x = 1

Solve equation b)
x + 6 = 7
x = 13
Check solutions:
 solution x = 1
Left Side of Equation for x = 1.
1 + 6 
=  7 
= 7
Right Side of Equation for x = 1.
7  x = 13
Left Side of Equation for x = 1.
13 + 6 
=  7 
= 7
Right Side of Equation for x = 1.
7
The solutions to the given equation are x = 1 and x = 13
Matched Exercise 1: Solve the equation
Example 2: Solve the equation
Solution to Example 2:

Given
2 x / 2 + 3   4 = 10

We first write the equation in the form  A  = B. Add 4 to both sides and group like terms
2x / 2 + 3  = 6

Divide both sides by 2
x / 2 + 3  = 3

We now proceed as in example 1 above, the equation
x / 2 + 3  = 3 gives two equations.
a)x / 2 + 3 = 3
or
b)x / 2 + 3 = 3

Solve equation a)
x / 2 + 3 = 3

to obtain
x = 0

Solve equation b)
x / 2 + 3 = 3

to obtain
x = 12
Check solutions:
 x = 0
Left Side of Equation for x = 0.
2 x / 2 + 3   4
= 2 3   4
= 10
Right Side of Equation for x = 1.
10  x = 12
Left Side of Equation for x = 12.
2 x / 2 + 3   4
= 2 12 / 2 + 3   4
= 2 6 + 3   4
= 2(3)  4
= 10
Right Side of Equation for x = 12.
10
The solutions to the given equation are x = 0 and x = 12
Matched Exercise 2: Solve the equation
Example 3: Solve the equation
Solution to Example 3:

If 2 x  2 >= 0 which is equivalent to x >= 1, then 2 x  2  = 2 x  2 and the given equation becomes
2 x  2 = x + 1

Add 2  x to both sides
x = 3
 Since x = 3 satisfies the condition x >= 1, it is a solution.

If 2x  2 < 0 which is equivalent to x < 1, then 2 x  2  = (2 x  2) and the given equation becomes
(2 x  2) = x + 1

Solve for x to obtain
x = 1 / 3
 Since x = 1 / 3 satisfies the condition x < 1, it is a solution.
Check solutions
 x = 3
Left Side of Equation for x = 3.
2 x  2 
= 2*3  2 
= 4
Right Side of Equation for x = 3.
x + 1
= 3 + 1
= 4  x = 1/3
Left Side of Equation for x = 1 / 3.
2 x  2 
= 2*(1/3)  2 
= 4 / 3
Right Side of Equation for x = 1 / 3.
x + 1
= 4 / 3
The solutions to the given equation are x = 3 and x = 1 / 3
Matched Exercise 3:Solve the equation
 4x + 2  = x  8
Example 4: Solve the equation
Solution to Example 3:

If x^{2}  4 >= 0 ,or x^{2} >= 4, then  x^{2}  4  = x^{2}  4 and the given equation becomes
x^{2}  4 = x + 2

Add  (x + 2) to both sides
x^{2}  4 ( x + 2) = 0

Factor the left term
(x  2)(x + 2) ( x + 2) = 0
(x + 2)(x  2 1) = 0
(x + 2)(x  3) = 0

Using the factor theorem, we can write two simpler equations
x + 2 = 0
or
x  3 = 0

Solve the above equations for x to find two values of x that make the left side of the equation equal to zero.
x = 2 and x = 3.

Both values satisfy the condition x^{2} >= 4 and are solutions to the given equation.
x = 2 and x = 3.

If x^{2}  4 < 0 ,or x^{2} < 4, then  x^{2}  4  = (x^{2}  4) and the given equation becomes.
(x^{2}  4) = x + 2
(x^{2}  4)  ( x + 2) = 0

Factor the left term.
(x  2)(x + 2)  ( x + 2) = 0
(x  2)(x + 2) + ( x + 2) = 0
(x  2)(x + 2) + ( x + 2) = 0
(x + 2)(x  2 + 1) = 0
(x + 2)(x  1) = 0

Two values make the left side of the above equation equal to zero
x = 2 and x = 1.

Only x = 1 satisfies the condition x^{2} < 4
Check solutions:

x = 2
Right Side of Equation =  x^{2}  4 
=  (2)^{2}  4  = 0
Left Side of Equation = x + 2 = 2 + 2 = 0

x = 3
Left Side of Equation =  x^{2}  4 
=  3^{2}  4 
=  5 
= 5 Right Side of Equation = x + 2 = 3 + 2 = 5 
x = 1
Left Side of Equation =  x^{2}  4 
=  1^{2}  4  =   3  = 3 Right Side of Equation = x + 2 = 1 + 2 = 3
Conclusion
The solutions to the given equation are x = 2, x = 1 and x = 3.
Matched Exercise 4: Solve the equation
Exercises.(see answers below)
Solve the following absolute value equations
a)  x  4  = 9
b)  x^{ 2} + 4  = 5
c)  x^{ 2}  9  = x + 3
d)  x + 1  = x  3
e)  x  = 2
Answers to Above Exercises.
a) 5 , 13
b) 1 , 1
c) 3 , 2 , 4
d) no real solutions
e) 2 , 2
More references and links on how to Solve Equations, Systems of Equations and Inequalities and Step by Step Solver for Equation With Absolute Value.
Linear ProgrammingNew ! Online Step by Step Calculus Calculators and SolversNew ! Factor Quadratic Expressions  Step by Step CalculatorNew ! Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers  Interactive HTML5 Math Web Apps for Mobile LearningNew !  Free Online Graph Plotter for All Devices
Home Page  HTML5 Math Applets for Mobile Learning  Math Formulas for Mobile Learning  Algebra Questions  Math Worksheets  Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests  GRE practice  GMAT practice Precalculus Tutorials  Precalculus Questions and Problems  Precalculus Applets  Equations, Systems and Inequalities  Online Calculators  Graphing  Trigonometry  Trigonometry Worsheets  Geometry Tutorials  Geometry Calculators  Geometry Worksheets  Calculus Tutorials  Calculus Questions  Calculus Worksheets  Applied Math  Antennas  Math Software  Elementary Statistics High School Math  Middle School Math  Primary Math
Math Videos From Analyzemath