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Solution - Absolute value equations

Exact form: =0,2
=0 , -2

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
|+1|=|x+1|
without the absolute value bars:

|x|=|y||+1|=|x+1|
x=+y(+1)=(x+1)
x=y(+1)=(x+1)
+x=y(+1)=(x+1)
x=y(+1)=(x+1)

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||+1|=|x+1|
x=+y , +x=y(+1)=(x+1)
x=y , x=y(+1)=(x+1)

2. Solve the two equations for

3 additional steps

(1)=(x+1)

Swap sides:

(x+1)=(1)

Subtract from both sides:

(x+1)-1=(1)-1

Simplify the arithmetic:

x=(1)-1

Simplify the arithmetic:

x=0

7 additional steps

(1)=-(x+1)

Expand the parentheses:

(1)=-x-1

Swap sides:

-x-1=(1)

Add to both sides:

(-x-1)+1=(1)+1

Simplify the arithmetic:

-x=(1)+1

Simplify the arithmetic:

x=2

Multiply both sides by :

-x·-1=2·-1

Remove the one(s):

x=2·-1

Simplify the arithmetic:

x=2

3. List the solutions

=0,2
(2 solution(s))

4. Graph

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