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Solution - Absolute value equations

Exact form: x=53,3
x=\frac{5}{3} , 3
Mixed number form: x=123,3
x=1\frac{2}{3} , 3
Decimal form: x=1.667,3
x=1.667 , 3
See steps

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

|x−1|+2|x−2|=0

Add −2|x−2| to both sides of the equation:

|x−1|+2|x−2|−2|x−2|=−2|x−2|

Simplify the arithmetic

|x−1|=−2|x−2|

2. 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
|x−1|=−2|x−2|
without the absolute value bars:

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

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

3. Solve the two equations for x

11 additional steps

(x-1)=-2·(x-2)

Expand the parentheses:

(x-1)=-2x-2·-2

Simplify the arithmetic:

(x-1)=-2x+4

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

3x-1=(-2x+4)+2x

Group like terms:

3x-1=(-2x+2x)+4

Simplify the arithmetic:

3x−1=4

Add to both sides:

(3x-1)+1=4+1

Simplify the arithmetic:

3x=4+1

Simplify the arithmetic:

3x=5

Divide both sides by :

(3x)3=53

Simplify the fraction:

x=53

15 additional steps

(x-1)=-2·(-(x-2))

Expand the parentheses:

(x-1)=-2·(-x+2)

(x-1)=-2·-x-2·2

Group like terms:

(x-1)=(-2·-1)x-2·2

Multiply the coefficients:

(x-1)=2x-2·2

Simplify the arithmetic:

(x-1)=2x-4

Subtract from both sides:

(x-1)-2x=(2x-4)-2x

Group like terms:

(x-2x)-1=(2x-4)-2x

Simplify the arithmetic:

-x-1=(2x-4)-2x

Group like terms:

-x-1=(2x-2x)-4

Simplify the arithmetic:

−x−1=−4

Add to both sides:

(-x-1)+1=-4+1

Simplify the arithmetic:

−x=−4+1

Simplify the arithmetic:

−x=−3

Multiply both sides by :

-x·-1=-3·-1

Remove the one(s):

x=-3·-1

Simplify the arithmetic:

x=3

4. List the solutions

x=53,3
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
y=|x−1|
y=−2|x−2|
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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