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Solution - Absolute value equations

Exact form:

x = \frac{32}{13}, \frac{16}{19}

x=\frac{32}{13} , \frac{16}{19}

Mixed number form:

x = 2 \frac{6}{13}, \frac{16}{19}

x=2\frac{6}{13} , \frac{16}{19}

Decimal form:

x = 2.462, 0.842

x=2.462 , 0.842

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
$| 3 x + 8 | = 8 | 2 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x + 8 \| = 8 \| 2 x - 3 \|$
$x = + y$	$(3 x + 8) = 8 (2 x - 3)$
$x = - y$	$(3 x + 8) = 8 (- (2 x - 3))$
$+ x = y$	$(3 x + 8) = 8 (2 x - 3)$
$- x = y$	$- (3 x + 8) = 8 (2 x - 3)$

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:

$\| x \| = \| y \|$	$\| 3 x + 8 \| = 8 \| 2 x - 3 \|$
$x = + y$ , $+ x = y$	$(3 x + 8) = 8 (2 x - 3)$
$x = - y$ , $- x = y$	$(3 x + 8) = 8 (- (2 x - 3))$

2. Solve the two equations for x

14 additional steps

$(3 x + 8) = 8 \cdot (2 x - 3)$

Expand the parentheses:

$(3 x + 8) = 8 \cdot 2 x + 8 \cdot - 3$

Multiply the coefficients:

$(3 x + 8) = 16 x + 8 \cdot - 3$

Simplify the arithmetic:

$(3 x + 8) = 16 x - 24$

Subtract from both sides:

$(3 x + 8) - 16 x = (16 x - 24) - 16 x$

Group like terms:

$(3 x - 16 x) + 8 = (16 x - 24) - 16 x$

Simplify the arithmetic:

$- 13 x + 8 = (16 x - 24) - 16 x$

Group like terms:

$- 13 x + 8 = (16 x - 16 x) - 24$

Simplify the arithmetic:

$- 13 x + 8 = - 24$

Subtract from both sides:

$(- 13 x + 8) - 8 = - 24 - 8$

Simplify the arithmetic:

$- 13 x = - 24 - 8$

Simplify the arithmetic:

$- 13 x = - 32$

Divide both sides by :

$\frac{(- 13 x)}{- 13} = \frac{- 32}{- 13}$

Cancel out the negatives:

$\frac{13 x}{13} = \frac{- 32}{- 13}$

Simplify the fraction:

$x = \frac{- 32}{- 13}$

Cancel out the negatives:

$x = \frac{32}{13}$

13 additional steps

$(3 x + 8) = 8 \cdot (- (2 x - 3))$

Expand the parentheses:

$(3 x + 8) = 8 \cdot (- 2 x + 3)$

Expand the parentheses:

$(3 x + 8) = 8 \cdot - 2 x + 8 \cdot 3$

Multiply the coefficients:

$(3 x + 8) = - 16 x + 8 \cdot 3$

Simplify the arithmetic:

$(3 x + 8) = - 16 x + 24$

Add to both sides:

$(3 x + 8) + 16 x = (- 16 x + 24) + 16 x$

Group like terms:

$(3 x + 16 x) + 8 = (- 16 x + 24) + 16 x$

Simplify the arithmetic:

$19 x + 8 = (- 16 x + 24) + 16 x$

Group like terms:

$19 x + 8 = (- 16 x + 16 x) + 24$

Simplify the arithmetic:

$19 x + 8 = 24$

Subtract from both sides:

$(19 x + 8) - 8 = 24 - 8$

Simplify the arithmetic:

$19 x = 24 - 8$

Simplify the arithmetic:

$19 x = 16$

Divide both sides by :

$\frac{(19 x)}{19} = \frac{16}{19}$

Simplify the fraction:

$x = \frac{16}{19}$

3. List the solutions

$x = \frac{32}{13}, \frac{16}{19}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 x + 8 |$
$y = 8 | 2 x - 3 |$
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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