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Solution - Absolute value equations

Exact form:

x = - 3, - 7

x=-3 , -7

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
$3 | x + 9 | = 6 | x + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| x + 9 \| = 6 \| x + 6 \|$
$x = + y$	$3 (x + 9) = 6 (x + 6)$
$x = - y$	$3 (x + 9) = 6 (- (x + 6))$
$+ x = y$	$3 (x + 9) = 6 (x + 6)$
$- x = y$	$3 (- (x + 9)) = 6 (x + 6)$

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:

$\| x \| = \| y \|$	$3 \| x + 9 \| = 6 \| x + 6 \|$
$x = + y$ , $+ x = y$	$3 (x + 9) = 6 (x + 6)$
$x = - y$ , $- x = y$	$3 (x + 9) = 6 (- (x + 6))$

2. Solve the two equations for x

17 additional steps

$3 \cdot (x + 9) = 6 \cdot (x + 6)$

Expand the parentheses:

$3 x + 3 \cdot 9 = 6 \cdot (x + 6)$

Simplify the arithmetic:

$3 x + 27 = 6 \cdot (x + 6)$

Expand the parentheses:

$3 x + 27 = 6 x + 6 \cdot 6$

Simplify the arithmetic:

$3 x + 27 = 6 x + 36$

Subtract from both sides:

$(3 x + 27) - 6 x = (6 x + 36) - 6 x$

Group like terms:

$(3 x - 6 x) + 27 = (6 x + 36) - 6 x$

Simplify the arithmetic:

$- 3 x + 27 = (6 x + 36) - 6 x$

Group like terms:

$- 3 x + 27 = (6 x - 6 x) + 36$

Simplify the arithmetic:

$- 3 x + 27 = 36$

Subtract from both sides:

$(- 3 x + 27) - 27 = 36 - 27$

Simplify the arithmetic:

$- 3 x = 36 - 27$

Simplify the arithmetic:

$- 3 x = 9$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{9}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{9}{- 3}$

Simplify the fraction:

$x = \frac{9}{- 3}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 9}{3}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = - 3$

18 additional steps

$3 \cdot (x + 9) = 6 \cdot (- (x + 6))$

Expand the parentheses:

$3 x + 3 \cdot 9 = 6 \cdot (- (x + 6))$

Simplify the arithmetic:

$3 x + 27 = 6 \cdot (- (x + 6))$

Expand the parentheses:

$3 x + 27 = 6 \cdot (- x - 6)$

$3 x + 27 = 6 \cdot - x + 6 \cdot - 6$

Group like terms:

$3 x + 27 = (6 \cdot - 1) x + 6 \cdot - 6$

Multiply the coefficients:

$3 x + 27 = - 6 x + 6 \cdot - 6$

Simplify the arithmetic:

$3 x + 27 = - 6 x - 36$

Add to both sides:

$(3 x + 27) + 6 x = (- 6 x - 36) + 6 x$

Group like terms:

$(3 x + 6 x) + 27 = (- 6 x - 36) + 6 x$

Simplify the arithmetic:

$9 x + 27 = (- 6 x - 36) + 6 x$

Group like terms:

$9 x + 27 = (- 6 x + 6 x) - 36$

Simplify the arithmetic:

$9 x + 27 = - 36$

Subtract from both sides:

$(9 x + 27) - 27 = - 36 - 27$

Simplify the arithmetic:

$9 x = - 36 - 27$

Simplify the arithmetic:

$9 x = - 63$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{- 63}{9}$

Simplify the fraction:

$x = \frac{- 63}{9}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 7 \cdot 9)}{(1 \cdot 9)}$

Factor out and cancel the greatest common factor:

$x = - 7$

3. List the solutions

$x = - 3, - 7$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | x + 9 |$
$y = 6 | x + 6 |$
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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