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Solution - Absolute value equations

Exact form: x=14
x=\frac{1}{4}
Decimal form: x=0.25
x=0.25

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
|2x1|=|2x|
without the absolute value bars:

|x|=|y||2x1|=|2x|
x=+y(2x1)=(2x)
x=y(2x1)=(2x)
+x=y(2x1)=(2x)
x=y(2x1)=(2x)

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||2x1|=|2x|
x=+y , +x=y(2x1)=(2x)
x=y , x=y(2x1)=(2x)

2. Solve the two equations for x

4 additional steps

(2x-1)=2x

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

-1=(2x)-2x

Simplify the arithmetic:

1=0

The statement is false:

1=0

The equation is false so it has no solution.

7 additional steps

(2x-1)=-2x

Add to both sides:

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

Simplify the arithmetic:

2x=(-2x)+1

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

4x=1

Divide both sides by :

(4x)4=14

Simplify the fraction:

x=14

3. Graph

Each line represents the function of one side of the equation:
y=|2x1|
y=|2x|
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.