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Solution - Absolute value equations

Exact form:

x = - 12, - \frac{12}{11}

x=-12 , -\frac{12}{11}

Mixed number form:

x = - 12, - 1 \frac{1}{11}

x=-12 , -1\frac{1}{11}

Decimal form:

x = - 12, - 1.091

x=-12 , -1.091

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
$5 | x | = 6 | x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| x \| = 6 \| x + 2 \|$
$x = + y$	$5 (x) = 6 (x + 2)$
$x = - y$	$5 (x) = 6 (- (x + 2))$
$+ x = y$	$5 (x) = 6 (x + 2)$
$- x = y$	$5 (- (x)) = 6 (x + 2)$

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:

$\| x \| = \| y \|$	$5 \| x \| = 6 \| x + 2 \|$
$x = + y$ , $+ x = y$	$5 (x) = 6 (x + 2)$
$x = - y$ , $- x = y$	$5 (x) = 6 (- (x + 2))$

2. Solve the two equations for x

8 additional steps

$5 x = 6 \cdot (x + 2)$

Expand the parentheses:

$5 x = 6 x + 6 \cdot 2$

Simplify the arithmetic:

$5 x = 6 x + 12$

Subtract from both sides:

$(5 x) - 6 x = (6 x + 12) - 6 x$

Simplify the arithmetic:

$- x = (6 x + 12) - 6 x$

Group like terms:

$- x = (6 x - 6 x) + 12$

Simplify the arithmetic:

$- x = 12$

Multiply both sides by :

$- x \cdot - 1 = 12 \cdot - 1$

Remove the one(s):

$x = 12 \cdot - 1$

Simplify the arithmetic:

$x = - 12$

10 additional steps

$5 x = 6 \cdot (- (x + 2))$

Expand the parentheses:

$5 x = 6 \cdot (- x - 2)$

$5 x = 6 \cdot - x + 6 \cdot - 2$

Group like terms:

$5 x = (6 \cdot - 1) x + 6 \cdot - 2$

Multiply the coefficients:

$5 x = - 6 x + 6 \cdot - 2$

Simplify the arithmetic:

$5 x = - 6 x - 12$

Add to both sides:

$(5 x) + 6 x = (- 6 x - 12) + 6 x$

Simplify the arithmetic:

$11 x = (- 6 x - 12) + 6 x$

Group like terms:

$11 x = (- 6 x + 6 x) - 12$

Simplify the arithmetic:

$11 x = - 12$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{- 12}{11}$

Simplify the fraction:

$x = \frac{- 12}{11}$

3. List the solutions

$x = - 12, - \frac{12}{11}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | x |$
$y = 6 | x + 2 |$
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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