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Solution - Absolute value equations

Exact form:

y = - \frac{45}{2}, \frac{45}{8}

y=-\frac{45}{2} , \frac{45}{8}

Mixed number form:

y = - 22 \frac{1}{2}, 5 \frac{5}{8}

y=-22\frac{1}{2} , 5\frac{5}{8}

Decimal form:

y = - 22.5, 5.625

y=-22.5 , 5.625

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
$5 | y | = 3 | y - 15 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| y \| = 3 \| y - 15 \|$
$x = + y$	$5 (y) = 3 (y - 15)$
$x = - y$	$5 (y) = 3 (- (y - 15))$
$+ x = y$	$5 (y) = 3 (y - 15)$
$- x = y$	$5 (- (y)) = 3 (y - 15)$

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:

$\| x \| = \| y \|$	$5 \| y \| = 3 \| y - 15 \|$
$x = + y$ , $+ x = y$	$5 (y) = 3 (y - 15)$
$x = - y$ , $- x = y$	$5 (y) = 3 (- (y - 15))$

2. Solve the two equations for y

7 additional steps

$5 y = 3 \cdot (y - 15)$

Expand the parentheses:

$5 y = 3 y + 3 \cdot - 15$

Simplify the arithmetic:

$5 y = 3 y - 45$

Subtract from both sides:

$(5 y) - 3 y = (3 y - 45) - 3 y$

Simplify the arithmetic:

$2 y = (3 y - 45) - 3 y$

Group like terms:

$2 y = (3 y - 3 y) - 45$

Simplify the arithmetic:

$2 y = - 45$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{- 45}{2}$

Simplify the fraction:

$y = \frac{- 45}{2}$

10 additional steps

$5 y = 3 \cdot (- (y - 15))$

Expand the parentheses:

$5 y = 3 \cdot (- y + 15)$

$5 y = 3 \cdot - y + 3 \cdot 15$

Group like terms:

$5 y = (3 \cdot - 1) y + 3 \cdot 15$

Multiply the coefficients:

$5 y = - 3 y + 3 \cdot 15$

Simplify the arithmetic:

$5 y = - 3 y + 45$

Add to both sides:

$(5 y) + 3 y = (- 3 y + 45) + 3 y$

Simplify the arithmetic:

$8 y = (- 3 y + 45) + 3 y$

Group like terms:

$8 y = (- 3 y + 3 y) + 45$

Simplify the arithmetic:

$8 y = 45$

Divide both sides by :

$\frac{(8 y)}{8} = \frac{45}{8}$

Simplify the fraction:

$y = \frac{45}{8}$

3. List the solutions

$y = - \frac{45}{2}, \frac{45}{8}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | y |$
$y = 3 | y - 15 |$
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 y

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 y

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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