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Solution - Absolute value equations

Exact form:

x = - \frac{20}{3}, \frac{8}{9}

x=-\frac{20}{3} , \frac{8}{9}

Mixed number form:

x = - 6 \frac{2}{3}, \frac{8}{9}

x=-6\frac{2}{3} , \frac{8}{9}

Decimal form:

x = - 6.667, 0.889

x=-6.667 , 0.889

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
$| 6 x + 6 | = | 3 x - 14 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 x + 6 \| = \| 3 x - 14 \|$
$x = + y$	$(6 x + 6) = (3 x - 14)$
$x = - y$	$(6 x + 6) = - (3 x - 14)$
$+ x = y$	$(6 x + 6) = (3 x - 14)$
$- x = y$	$- (6 x + 6) = (3 x - 14)$

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:

$\| x \| = \| y \|$	$\| 6 x + 6 \| = \| 3 x - 14 \|$
$x = + y$ , $+ x = y$	$(6 x + 6) = (3 x - 14)$
$x = - y$ , $- x = y$	$(6 x + 6) = - (3 x - 14)$

2. Solve the two equations for x

9 additional steps

$(6 x + 6) = (3 x - 14)$

Subtract from both sides:

$(6 x + 6) - 3 x = (3 x - 14) - 3 x$

Group like terms:

$(6 x - 3 x) + 6 = (3 x - 14) - 3 x$

Simplify the arithmetic:

$3 x + 6 = (3 x - 14) - 3 x$

Group like terms:

$3 x + 6 = (3 x - 3 x) - 14$

Simplify the arithmetic:

$3 x + 6 = - 14$

Subtract from both sides:

$(3 x + 6) - 6 = - 14 - 6$

Simplify the arithmetic:

$3 x = - 14 - 6$

Simplify the arithmetic:

$3 x = - 20$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{- 20}{3}$

Simplify the fraction:

$x = \frac{- 20}{3}$

10 additional steps

$(6 x + 6) = - (3 x - 14)$

Expand the parentheses:

$(6 x + 6) = - 3 x + 14$

Add to both sides:

$(6 x + 6) + 3 x = (- 3 x + 14) + 3 x$

Group like terms:

$(6 x + 3 x) + 6 = (- 3 x + 14) + 3 x$

Simplify the arithmetic:

$9 x + 6 = (- 3 x + 14) + 3 x$

Group like terms:

$9 x + 6 = (- 3 x + 3 x) + 14$

Simplify the arithmetic:

$9 x + 6 = 14$

Subtract from both sides:

$(9 x + 6) - 6 = 14 - 6$

Simplify the arithmetic:

$9 x = 14 - 6$

Simplify the arithmetic:

$9 x = 8$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{8}{9}$

Simplify the fraction:

$x = \frac{8}{9}$

3. List the solutions

$x = - \frac{20}{3}, \frac{8}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 x + 6 |$
$y = | 3 x - 14 |$
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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