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Solution - Absolute value equations

Exact form: z=5
z=-5

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+6|=|z+4|
without the absolute value bars:

|x|=|y||z+6|=|z+4|
x=+y(z+6)=(z+4)
x=y(z+6)=(z+4)
+x=y(z+6)=(z+4)
x=y(z+6)=(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+6|=|z+4|
x=+y , +x=y(z+6)=(z+4)
x=y , x=y(z+6)=(z+4)

2. Solve the two equations for z

5 additional steps

(z+6)=(z+4)

Subtract from both sides:

(z+6)-z=(z+4)-z

Group like terms:

(z-z)+6=(z+4)-z

Simplify the arithmetic:

6=(z+4)-z

Group like terms:

6=(z-z)+4

Simplify the arithmetic:

6=4

The statement is false:

6=4

The equation is false so it has no solution.

12 additional steps

(z+6)=-(z+4)

Expand the parentheses:

(z+6)=-z-4

Add to both sides:

(z+6)+z=(-z-4)+z

Group like terms:

(z+z)+6=(-z-4)+z

Simplify the arithmetic:

2z+6=(-z-4)+z

Group like terms:

2z+6=(-z+z)-4

Simplify the arithmetic:

2z+6=4

Subtract from both sides:

(2z+6)-6=-4-6

Simplify the arithmetic:

2z=46

Simplify the arithmetic:

2z=10

Divide both sides by :

(2z)2=-102

Simplify the fraction:

z=-102

Find the greatest common factor of the numerator and denominator:

z=(-5·2)(1·2)

Factor out and cancel the greatest common factor:

z=5

3. Graph

Each line represents the function of one side of the equation:
y=|z+6|
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.