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Solution - Absolute value equations

Exact form:

x = - 3, \frac{3}{2}

x=-3 , \frac{3}{2}

Mixed number form:

x = - 3, 1 \frac{1}{2}

x=-3 , 1\frac{1}{2}

Decimal form:

x = - 3, 1.5

x=-3 , 1.5

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| x - 6 | = 3 | x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 6 \| = 3 \| x \|$
$x = + y$	$(x - 6) = 3 (x)$
$x = - y$	$(x - 6) = 3 (- (x))$
$+ x = y$	$(x - 6) = 3 (x)$
$- x = y$	$- (x - 6) = 3 (x)$

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:

$\| x \| = \| y \|$	$\| x - 6 \| = 3 \| x \|$
$x = + y$ , $+ x = y$	$(x - 6) = 3 (x)$
$x = - y$ , $- x = y$	$(x - 6) = 3 (- (x))$

2. Solve the two equations for x

12 additional steps

$(x - 6) = 3 x$

Subtract from both sides:

$(x - 6) - 3 x = (3 x) - 3 x$

Group like terms:

$(x - 3 x) - 6 = (3 x) - 3 x$

Simplify the arithmetic:

$- 2 x - 6 = (3 x) - 3 x$

Simplify the arithmetic:

$- 2 x - 6 = 0$

Add to both sides:

$(- 2 x - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$- 2 x = 0 + 6$

Simplify the arithmetic:

$- 2 x = 6$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{6}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{6}{- 2}$

Simplify the fraction:

$x = \frac{6}{- 2}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 6}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 3 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = - 3$

12 additional steps

$(x - 6) = 3 \cdot - x$

Group like terms:

$(x - 6) = (3 \cdot - 1) x$

Multiply the coefficients:

$(x - 6) = - 3 x$

Add to both sides:

$(x - 6) + 3 x = (- 3 x) + 3 x$

Group like terms:

$(x + 3 x) - 6 = (- 3 x) + 3 x$

Simplify the arithmetic:

$4 x - 6 = (- 3 x) + 3 x$

Simplify the arithmetic:

$4 x - 6 = 0$

Add to both sides:

$(4 x - 6) + 6 = 0 + 6$

Simplify the arithmetic:

$4 x = 0 + 6$

Simplify the arithmetic:

$4 x = 6$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{6}{4}$

Simplify the fraction:

$x = \frac{6}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{3}{2}$

3. List the solutions

$x = - 3, \frac{3}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 6 |$
$y = 3 | x |$
The equation is true where the two lines cross.

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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.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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