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Solution - Absolute value equations

Exact form:

x = - 3, - 3

x=-3 , -3

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
$2 | x + 3 | = 3 | 2 x + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| x + 3 \| = 3 \| 2 x + 6 \|$
$x = + y$	$2 (x + 3) = 3 (2 x + 6)$
$x = - y$	$2 (x + 3) = 3 (- (2 x + 6))$
$+ x = y$	$2 (x + 3) = 3 (2 x + 6)$
$- x = y$	$2 (- (x + 3)) = 3 (2 x + 6)$

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:

$\| x \| = \| y \|$	$2 \| x + 3 \| = 3 \| 2 x + 6 \|$
$x = + y$ , $+ x = y$	$2 (x + 3) = 3 (2 x + 6)$
$x = - y$ , $- x = y$	$2 (x + 3) = 3 (- (2 x + 6))$

2. Solve the two equations for x

18 additional steps

$2 \cdot (x + 3) = 3 \cdot (2 x + 6)$

Expand the parentheses:

$2 x + 2 \cdot 3 = 3 \cdot (2 x + 6)$

Simplify the arithmetic:

$2 x + 6 = 3 \cdot (2 x + 6)$

Expand the parentheses:

$2 x + 6 = 3 \cdot 2 x + 3 \cdot 6$

Multiply the coefficients:

$2 x + 6 = 6 x + 3 \cdot 6$

Simplify the arithmetic:

$2 x + 6 = 6 x + 18$

Subtract from both sides:

$(2 x + 6) - 6 x = (6 x + 18) - 6 x$

Group like terms:

$(2 x - 6 x) + 6 = (6 x + 18) - 6 x$

Simplify the arithmetic:

$- 4 x + 6 = (6 x + 18) - 6 x$

Group like terms:

$- 4 x + 6 = (6 x - 6 x) + 18$

Simplify the arithmetic:

$- 4 x + 6 = 18$

Subtract from both sides:

$(- 4 x + 6) - 6 = 18 - 6$

Simplify the arithmetic:

$- 4 x = 18 - 6$

Simplify the arithmetic:

$- 4 x = 12$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{12}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{12}{- 4}$

Simplify the fraction:

$x = \frac{12}{- 4}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 12}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 3 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = - 3$

17 additional steps

$2 \cdot (x + 3) = 3 \cdot (- (2 x + 6))$

Expand the parentheses:

$2 x + 2 \cdot 3 = 3 \cdot (- (2 x + 6))$

Simplify the arithmetic:

$2 x + 6 = 3 \cdot (- (2 x + 6))$

Expand the parentheses:

$2 x + 6 = 3 \cdot (- 2 x - 6)$

Expand the parentheses:

$2 x + 6 = 3 \cdot - 2 x + 3 \cdot - 6$

Multiply the coefficients:

$2 x + 6 = - 6 x + 3 \cdot - 6$

Simplify the arithmetic:

$2 x + 6 = - 6 x - 18$

Add to both sides:

$(2 x + 6) + 6 x = (- 6 x - 18) + 6 x$

Group like terms:

$(2 x + 6 x) + 6 = (- 6 x - 18) + 6 x$

Simplify the arithmetic:

$8 x + 6 = (- 6 x - 18) + 6 x$

Group like terms:

$8 x + 6 = (- 6 x + 6 x) - 18$

Simplify the arithmetic:

$8 x + 6 = - 18$

Subtract from both sides:

$(8 x + 6) - 6 = - 18 - 6$

Simplify the arithmetic:

$8 x = - 18 - 6$

Simplify the arithmetic:

$8 x = - 24$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{- 24}{8}$

Simplify the fraction:

$x = \frac{- 24}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 3 \cdot 8)}{(1 \cdot 8)}$

Factor out and cancel the greatest common factor:

$x = - 3$

3. List the solutions

$x = - 3, - 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | x + 3 |$
$y = 3 | 2 x + 6 |$
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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