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Solution - Absolute value equations

Exact form:

x = - \frac{16}{5}, - \frac{8}{13}

x=-\frac{16}{5} , -\frac{8}{13}

Mixed number form:

x = - 3 \frac{1}{5}, - \frac{8}{13}

x=-3\frac{1}{5} , -\frac{8}{13}

Decimal form:

x = - 3.2, - 0.615

x=-3.2 , -0.615

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
$| 4 x - 4 | = | 9 x + 12 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 4 \| = \| 9 x + 12 \|$
$x = + y$	$(4 x - 4) = (9 x + 12)$
$x = - y$	$(4 x - 4) = - (9 x + 12)$
$+ x = y$	$(4 x - 4) = (9 x + 12)$
$- x = y$	$- (4 x - 4) = (9 x + 12)$

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:

$\| x \| = \| y \|$	$\| 4 x - 4 \| = \| 9 x + 12 \|$
$x = + y$ , $+ x = y$	$(4 x - 4) = (9 x + 12)$
$x = - y$ , $- x = y$	$(4 x - 4) = - (9 x + 12)$

2. Solve the two equations for x

11 additional steps

$(4 x - 4) = (9 x + 12)$

Subtract from both sides:

$(4 x - 4) - 9 x = (9 x + 12) - 9 x$

Group like terms:

$(4 x - 9 x) - 4 = (9 x + 12) - 9 x$

Simplify the arithmetic:

$- 5 x - 4 = (9 x + 12) - 9 x$

Group like terms:

$- 5 x - 4 = (9 x - 9 x) + 12$

Simplify the arithmetic:

$- 5 x - 4 = 12$

Add to both sides:

$(- 5 x - 4) + 4 = 12 + 4$

Simplify the arithmetic:

$- 5 x = 12 + 4$

Simplify the arithmetic:

$- 5 x = 16$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{16}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{16}{- 5}$

Simplify the fraction:

$x = \frac{16}{- 5}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 16}{5}$

10 additional steps

$(4 x - 4) = - (9 x + 12)$

Expand the parentheses:

$(4 x - 4) = - 9 x - 12$

Add to both sides:

$(4 x - 4) + 9 x = (- 9 x - 12) + 9 x$

Group like terms:

$(4 x + 9 x) - 4 = (- 9 x - 12) + 9 x$

Simplify the arithmetic:

$13 x - 4 = (- 9 x - 12) + 9 x$

Group like terms:

$13 x - 4 = (- 9 x + 9 x) - 12$

Simplify the arithmetic:

$13 x - 4 = - 12$

Add to both sides:

$(13 x - 4) + 4 = - 12 + 4$

Simplify the arithmetic:

$13 x = - 12 + 4$

Simplify the arithmetic:

$13 x = - 8$

Divide both sides by :

$\frac{(13 x)}{13} = \frac{- 8}{13}$

Simplify the fraction:

$x = \frac{- 8}{13}$

3. List the solutions

$x = - \frac{16}{5}, - \frac{8}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 4 |$
$y = | 9 x + 12 |$
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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