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Solution - Absolute value equations

Exact form: z=32
z=\frac{3}{2}
Mixed number form: z=112
z=1\frac{1}{2}
Decimal form: z=1.5
z=1.5
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+1|=|z−4|
without the absolute value bars:

|x|=|y||z+1|=|z−4|
x=+y(z+1)=(z−4)
x=−y(z+1)=−(z−4)
+x=y(z+1)=(z−4)
−x=y−(z+1)=(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+1|=|z−4|
x=+y , +x=y(z+1)=(z−4)
x=−y , −x=y(z+1)=−(z−4)

2. Solve the two equations for z

5 additional steps

(z+1)=(z-4)

Subtract from both sides:

(z+1)-z=(z-4)-z

Group like terms:

(z-z)+1=(z-4)-z

Simplify the arithmetic:

1=(z-4)-z

Group like terms:

1=(z-z)-4

Simplify the arithmetic:

1=−4

The statement is false:

1=−4

The equation is false so it has no solution.

10 additional steps

(z+1)=-(z-4)

Expand the parentheses:

(z+1)=-z+4

Add to both sides:

(z+1)+z=(-z+4)+z

Group like terms:

(z+z)+1=(-z+4)+z

Simplify the arithmetic:

2z+1=(-z+4)+z

Group like terms:

2z+1=(-z+z)+4

Simplify the arithmetic:

2z+1=4

Subtract from both sides:

(2z+1)-1=4-1

Simplify the arithmetic:

2z=4−1

Simplify the arithmetic:

2z=3

Divide both sides by :

(2z)2=32

Simplify the fraction:

z=32

3. Graph

Each line represents the function of one side of the equation:
y=|z+1|
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.

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