Copyright Ⓒ 2013-2025
tiger-algebra.com

☰
Tiger Algebra Calculator Formatting help
Select Language
Add to Home
Tiger Algebra Solver
Are you looking for...
Camera Icon
Enter an equation or problem
Camera Icon
Camera input is not recognized!

Solution - Absolute value equations

Exact form: z=−2
z=-2
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
|x|=|y| → x=±y and |x|=|y| → ±x=y
to write all four options of the equation
|z+8|=|z−4|
without the absolute value bars:

|x|=|y||z+8|=|z−4|
x=+y(z+8)=(z−4)
x=−y(z+8)=−(z−4)
+x=y(z+8)=(z−4)
−x=y−(z+8)=(z−4)

When simplified, equations x=+y and +x=y are the same and equations x=−y and −x=y are the same, so we end up with only 2 equations:

|x|=|y||z+8|=|z−4|
x=+y , +x=y(z+8)=(z−4)
x=−y , −x=y(z+8)=−(z−4)

2. Solve the two equations for z

5 additional steps

(z+8)=(z-4)

Subtract from both sides:

(z+8)-z=(z-4)-z

Group like terms:

(z-z)+8=(z-4)-z

Simplify the arithmetic:

8=(z-4)-z

Group like terms:

8=(z-z)-4

Simplify the arithmetic:

8=−4

The statement is false:

8=−4

The equation is false so it has no solution.

12 additional steps

(z+8)=-(z-4)

Expand the parentheses:

(z+8)=-z+4

Add to both sides:

(z+8)+z=(-z+4)+z

Group like terms:

(z+z)+8=(-z+4)+z

Simplify the arithmetic:

2z+8=(-z+4)+z

Group like terms:

2z+8=(-z+z)+4

Simplify the arithmetic:

2z+8=4

Subtract from both sides:

(2z+8)-8=4-8

Simplify the arithmetic:

2z=4−8

Simplify the arithmetic:

2z=−4

Divide both sides by :

(2z)2=-42

Simplify the fraction:

z=-42

Find the greatest common factor of the numerator and denominator:

z=(-2·2)(1·2)

Factor out and cancel the greatest common factor:

z=−2

3. Graph

Each line represents the function of one side of the equation:
y=|z+8|
y=|z−4|
The equation is true where the two lines cross.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Latest Related Drills Solved