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Solution - Absolute value equations

Exact form:

x = 6, - \frac{12}{7}

x=6 , -\frac{12}{7}

Mixed number form:

x = 6, - 1 \frac{5}{7}

x=6 , -1\frac{5}{7}

Decimal form:

x = 6, - 1.714

x=6 , -1.714

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
$6 | x + 3 | = 2 | 4 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$6 \| x + 3 \| = 2 \| 4 x + 3 \|$
$x = + y$	$6 (x + 3) = 2 (4 x + 3)$
$x = - y$	$6 (x + 3) = 2 (- (4 x + 3))$
$+ x = y$	$6 (x + 3) = 2 (4 x + 3)$
$- x = y$	$6 (- (x + 3)) = 2 (4 x + 3)$

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:

$\| x \| = \| y \|$	$6 \| x + 3 \| = 2 \| 4 x + 3 \|$
$x = + y$ , $+ x = y$	$6 (x + 3) = 2 (4 x + 3)$
$x = - y$ , $- x = y$	$6 (x + 3) = 2 (- (4 x + 3))$

2. Solve the two equations for x

18 additional steps

$6 \cdot (x + 3) = 2 \cdot (4 x + 3)$

Expand the parentheses:

$6 x + 6 \cdot 3 = 2 \cdot (4 x + 3)$

Simplify the arithmetic:

$6 x + 18 = 2 \cdot (4 x + 3)$

Expand the parentheses:

$6 x + 18 = 2 \cdot 4 x + 2 \cdot 3$

Multiply the coefficients:

$6 x + 18 = 8 x + 2 \cdot 3$

Simplify the arithmetic:

$6 x + 18 = 8 x + 6$

Subtract from both sides:

$(6 x + 18) - 8 x = (8 x + 6) - 8 x$

Group like terms:

$(6 x - 8 x) + 18 = (8 x + 6) - 8 x$

Simplify the arithmetic:

$- 2 x + 18 = (8 x + 6) - 8 x$

Group like terms:

$- 2 x + 18 = (8 x - 8 x) + 6$

Simplify the arithmetic:

$- 2 x + 18 = 6$

Subtract from both sides:

$(- 2 x + 18) - 18 = 6 - 18$

Simplify the arithmetic:

$- 2 x = 6 - 18$

Simplify the arithmetic:

$- 2 x = - 12$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{- 12}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{- 12}{- 2}$

Simplify the fraction:

$x = \frac{- 12}{- 2}$

Cancel out the negatives:

$x = \frac{12}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(6 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 6$

17 additional steps

$6 \cdot (x + 3) = 2 \cdot (- (4 x + 3))$

Expand the parentheses:

$6 x + 6 \cdot 3 = 2 \cdot (- (4 x + 3))$

Simplify the arithmetic:

$6 x + 18 = 2 \cdot (- (4 x + 3))$

Expand the parentheses:

$6 x + 18 = 2 \cdot (- 4 x - 3)$

Expand the parentheses:

$6 x + 18 = 2 \cdot - 4 x + 2 \cdot - 3$

Multiply the coefficients:

$6 x + 18 = - 8 x + 2 \cdot - 3$

Simplify the arithmetic:

$6 x + 18 = - 8 x - 6$

Add to both sides:

$(6 x + 18) + 8 x = (- 8 x - 6) + 8 x$

Group like terms:

$(6 x + 8 x) + 18 = (- 8 x - 6) + 8 x$

Simplify the arithmetic:

$14 x + 18 = (- 8 x - 6) + 8 x$

Group like terms:

$14 x + 18 = (- 8 x + 8 x) - 6$

Simplify the arithmetic:

$14 x + 18 = - 6$

Subtract from both sides:

$(14 x + 18) - 18 = - 6 - 18$

Simplify the arithmetic:

$14 x = - 6 - 18$

Simplify the arithmetic:

$14 x = - 24$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{- 24}{14}$

Simplify the fraction:

$x = \frac{- 24}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 12 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 12}{7}$

3. List the solutions

$x = 6, - \frac{12}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 6 | x + 3 |$
$y = 2 | 4 x + 3 |$
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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