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Solution - Absolute value equations

Exact form:

x = - \frac{27}{7}, - \frac{3}{13}

x=-\frac{27}{7} , -\frac{3}{13}

Mixed number form:

x = - 3 \frac{6}{7}, - \frac{3}{13}

x=-3\frac{6}{7} , -\frac{3}{13}

Decimal form:

x = - 3.857, - 0.231

x=-3.857 , -0.231

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
$3 | x - 4 | = 5 | 2 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| x - 4 \| = 5 \| 2 x + 3 \|$
$x = + y$	$3 (x - 4) = 5 (2 x + 3)$
$x = - y$	$3 (x - 4) = 5 (- (2 x + 3))$
$+ x = y$	$3 (x - 4) = 5 (2 x + 3)$
$- x = y$	$3 (- (x - 4)) = 5 (2 x + 3)$

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:

$\| x \| = \| y \|$	$3 \| x - 4 \| = 5 \| 2 x + 3 \|$
$x = + y$ , $+ x = y$	$3 (x - 4) = 5 (2 x + 3)$
$x = - y$ , $- x = y$	$3 (x - 4) = 5 (- (2 x + 3))$

2. Solve the two equations for x

16 additional steps

$3 \cdot (x - 4) = 5 \cdot (2 x + 3)$

Expand the parentheses:

$3 x + 3 \cdot - 4 = 5 \cdot (2 x + 3)$

Simplify the arithmetic:

$3 x - 12 = 5 \cdot (2 x + 3)$

Expand the parentheses:

$3 x - 12 = 5 \cdot 2 x + 5 \cdot 3$

Multiply the coefficients:

$3 x - 12 = 10 x + 5 \cdot 3$

Simplify the arithmetic:

$3 x - 12 = 10 x + 15$

Subtract from both sides:

$(3 x - 12) - 10 x = (10 x + 15) - 10 x$

Group like terms:

$(3 x - 10 x) - 12 = (10 x + 15) - 10 x$

Simplify the arithmetic:

$- 7 x - 12 = (10 x + 15) - 10 x$

Group like terms:

$- 7 x - 12 = (10 x - 10 x) + 15$

Simplify the arithmetic:

$- 7 x - 12 = 15$

Add to both sides:

$(- 7 x - 12) + 12 = 15 + 12$

Simplify the arithmetic:

$- 7 x = 15 + 12$

Simplify the arithmetic:

$- 7 x = 27$

Divide both sides by :

$\frac{(- 7 x)}{- 7} = \frac{27}{- 7}$

Cancel out the negatives:

$\frac{7 x}{7} = \frac{27}{- 7}$

Simplify the fraction:

$x = \frac{27}{- 7}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 27}{7}$

15 additional steps

$3 \cdot (x - 4) = 5 \cdot (- (2 x + 3))$

Expand the parentheses:

$3 x + 3 \cdot - 4 = 5 \cdot (- (2 x + 3))$

Simplify the arithmetic:

$3 x - 12 = 5 \cdot (- (2 x + 3))$

Expand the parentheses:

$3 x - 12 = 5 \cdot (- 2 x - 3)$

Expand the parentheses:

$3 x - 12 = 5 \cdot - 2 x + 5 \cdot - 3$

Multiply the coefficients:

$3 x - 12 = - 10 x + 5 \cdot - 3$

Simplify the arithmetic:

$3 x - 12 = - 10 x - 15$

Add to both sides:

$(3 x - 12) + 10 x = (- 10 x - 15) + 10 x$

Group like terms:

$(3 x + 10 x) - 12 = (- 10 x - 15) + 10 x$

Simplify the arithmetic:

$13 x - 12 = (- 10 x - 15) + 10 x$

Group like terms:

$13 x - 12 = (- 10 x + 10 x) - 15$

Simplify the arithmetic:

$13 x - 12 = - 15$

Add to both sides:

$(13 x - 12) + 12 = - 15 + 12$

Simplify the arithmetic:

$13 x = - 15 + 12$

Simplify the arithmetic:

$13 x = - 3$

Divide both sides by :

$\frac{(13 x)}{13} = \frac{- 3}{13}$

Simplify the fraction:

$x = \frac{- 3}{13}$

3. List the solutions

$x = - \frac{27}{7}, - \frac{3}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | x - 4 |$
$y = 5 | 2 x + 3 |$
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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