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Solution - Absolute value equations

Exact form: z=2,1
z=-2 , 1

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
3|z|=|z4|
without the absolute value bars:

|x|=|y|3|z|=|z4|
x=+y3(z)=(z4)
x=y3(z)=(z4)
+x=y3(z)=(z4)
x=y3((z))=(z4)

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|3|z|=|z4|
x=+y , +x=y3(z)=(z4)
x=y , x=y3(z)=(z4)

2. Solve the two equations for z

7 additional steps

3z=(z-4)

Subtract from both sides:

(3z)-z=(z-4)-z

Simplify the arithmetic:

2z=(z-4)-z

Group like terms:

2z=(z-z)-4

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

7 additional steps

3z=-(z-4)

Expand the parentheses:

3z=z+4

Add to both sides:

(3z)+z=(-z+4)+z

Simplify the arithmetic:

4z=(-z+4)+z

Group like terms:

4z=(-z+z)+4

Simplify the arithmetic:

4z=4

Divide both sides by :

(4z)4=44

Simplify the fraction:

z=44

Simplify the fraction:

z=1

3. List the solutions

z=2,1
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=3|z|
y=|z4|
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.