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Solution - Absolute value equations

Exact form: x=1,-13
x=1 , -\frac{1}{3}
Decimal form: x=1,−0.333
x=1 , -0.333
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
|2x|=|x+1|
without the absolute value bars:

|x|=|y||2x|=|x+1|
x=+y(2x)=(x+1)
x=−y(2x)=−(x+1)
+x=y(2x)=(x+1)
−x=y−(2x)=(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||2x|=|x+1|
x=+y , +x=y(2x)=(x+1)
x=−y , −x=y(2x)=−(x+1)

2. Solve the two equations for x

3 additional steps

2x=(x+1)

Subtract from both sides:

(2x)-x=(x+1)-x

Simplify the arithmetic:

x=(x+1)-x

Group like terms:

x=(x-x)+1

Simplify the arithmetic:

x=1

6 additional steps

2x=-(x+1)

Expand the parentheses:

2x=−x−1

Add to both sides:

(2x)+x=(-x-1)+x

Simplify the arithmetic:

3x=(-x-1)+x

Group like terms:

3x=(-x+x)-1

Simplify the arithmetic:

3x=−1

Divide both sides by :

(3x)3=-13

Simplify the fraction:

x=-13

3. List the solutions

x=1,-13
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2x|
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.

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