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Solution - Absolute value equations

Exact form:

x = - \frac{7}{3}, - 17

x=-\frac{7}{3} , -17

Mixed number form:

x = - 2 \frac{1}{3}, - 17

x=-2\frac{1}{3} , -17

Decimal form:

x = - 2.333, - 17

x=-2.333 , -17

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

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

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:

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

2. Solve the two equations for x

11 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

$2 x + 12 = (- x + 5)$

Add to both sides:

$(2 x + 12) + x = (- x + 5) + x$

Group like terms:

$(2 x + x) + 12 = (- x + 5) + x$

Simplify the arithmetic:

$3 x + 12 = (- x + 5) + x$

Group like terms:

$3 x + 12 = (- x + x) + 5$

Simplify the arithmetic:

$3 x + 12 = 5$

Subtract from both sides:

$(3 x + 12) - 12 = 5 - 12$

Simplify the arithmetic:

$3 x = 5 - 12$

Simplify the arithmetic:

$3 x = - 7$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{- 7}{3}$

Simplify the fraction:

$x = \frac{- 7}{3}$

10 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

$2 x + 12 = - (- x + 5)$

Expand the parentheses:

$2 x + 12 = x - 5$

Subtract from both sides:

$(2 x + 12) - x = (x - 5) - x$

Group like terms:

$(2 x - x) + 12 = (x - 5) - x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$x + 12 = - 5$

Subtract from both sides:

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

Simplify the arithmetic:

$x = - 5 - 12$

Simplify the arithmetic:

$x = - 17$

3. List the solutions

$x = - \frac{7}{3}, - 17$
(2 solution(s))

4. Graph

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