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Solution - Absolute value equations

Exact form:

x = - 18, - \frac{18}{17}

x=-18 , -\frac{18}{17}

Mixed number form:

x = - 18, - 1 \frac{1}{17}

x=-18 , -1\frac{1}{17}

Decimal form:

x = - 18, - 1.059

x=-18 , -1.059

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
$9 | x + 2 | = 8 | x |$
without the absolute value bars:

$\| x \| = \| y \|$	$9 \| x + 2 \| = 8 \| x \|$
$x = + y$	$9 (x + 2) = 8 (x)$
$x = - y$	$9 (x + 2) = 8 (- (x))$
$+ x = y$	$9 (x + 2) = 8 (x)$
$- x = y$	$9 (- (x + 2)) = 8 (x)$

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:

$\| x \| = \| y \|$	$9 \| x + 2 \| = 8 \| x \|$
$x = + y$ , $+ x = y$	$9 (x + 2) = 8 (x)$
$x = - y$ , $- x = y$	$9 (x + 2) = 8 (- (x))$

2. Solve the two equations for x

8 additional steps

$9 \cdot (x + 2) = 8 x$

Expand the parentheses:

$9 x + 9 \cdot 2 = 8 x$

Simplify the arithmetic:

$9 x + 18 = 8 x$

Subtract from both sides:

$(9 x + 18) - 8 x = (8 x) - 8 x$

Group like terms:

$(9 x - 8 x) + 18 = (8 x) - 8 x$

Simplify the arithmetic:

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

Simplify the arithmetic:

$x + 18 = 0$

Subtract from both sides:

$(x + 18) - 18 = 0 - 18$

Simplify the arithmetic:

$x = 0 - 18$

Simplify the arithmetic:

$x = - 18$

12 additional steps

$9 \cdot (x + 2) = 8 \cdot - x$

Expand the parentheses:

$9 x + 9 \cdot 2 = 8 \cdot - x$

Simplify the arithmetic:

$9 x + 18 = 8 \cdot - x$

Group like terms:

$9 x + 18 = (8 \cdot - 1) x$

Multiply the coefficients:

$9 x + 18 = - 8 x$

Add to both sides:

$(9 x + 18) + 8 x = (- 8 x) + 8 x$

Group like terms:

$(9 x + 8 x) + 18 = (- 8 x) + 8 x$

Simplify the arithmetic:

$17 x + 18 = (- 8 x) + 8 x$

Simplify the arithmetic:

$17 x + 18 = 0$

Subtract from both sides:

$(17 x + 18) - 18 = 0 - 18$

Simplify the arithmetic:

$17 x = 0 - 18$

Simplify the arithmetic:

$17 x = - 18$

Divide both sides by :

$\frac{(17 x)}{17} = \frac{- 18}{17}$

Simplify the fraction:

$x = \frac{- 18}{17}$

3. List the solutions

$x = - 18, - \frac{18}{17}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 9 | x + 2 |$
$y = 8 | x |$
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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