Copyright Ⓒ 2013-2025
tiger-algebra.com

☰
Camera input is not recognized!

Solution - Absolute value equations

Exact form: w=16
w=\frac{1}{6}
Decimal form: w=0.167
w=0.167
See steps

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
|6w−2|=|6w|
without the absolute value bars:

|x|=|y||6w−2|=|6w|
x=+y(6w−2)=(6w)
x=−y(6w−2)=−(6w)
+x=y(6w−2)=(6w)
−x=y−(6w−2)=(6w)

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||6w−2|=|6w|
x=+y , +x=y(6w−2)=(6w)
x=−y , −x=y(6w−2)=−(6w)

2. Solve the two equations for w

4 additional steps

(6w-2)=6w

Subtract from both sides:

(6w-2)-6w=(6w)-6w

Group like terms:

(6w-6w)-2=(6w)-6w

Simplify the arithmetic:

-2=(6w)-6w

Simplify the arithmetic:

−2=0

The statement is false:

−2=0

The equation is false so it has no solution.

9 additional steps

(6w-2)=-6w

Add to both sides:

(6w-2)+2=(-6w)+2

Simplify the arithmetic:

6w=(-6w)+2

Add to both sides:

(6w)+6w=((-6w)+2)+6w

Simplify the arithmetic:

12w=((-6w)+2)+6w

Group like terms:

12w=(-6w+6w)+2

Simplify the arithmetic:

12w=2

Divide both sides by :

(12w)12=212

Simplify the fraction:

w=212

Find the greatest common factor of the numerator and denominator:

w=(1·2)(6·2)

Factor out and cancel the greatest common factor:

w=16

3. Graph

Each line represents the function of one side of the equation:
y=|6w−2|
y=|6w|
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.

Latest Related Drills Solved