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Solution - Absolute value equations

Exact form:

y = - \frac{1}{2}, \frac{5}{4}

y=-\frac{1}{2} , \frac{5}{4}

Mixed number form:

y = - \frac{1}{2}, 1 \frac{1}{4}

y=-\frac{1}{2} , 1\frac{1}{4}

Decimal form:

y = - 0.5, 1.25

y=-0.5 , 1.25

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

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

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:

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

2. Solve the two equations for y

9 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 y + 3 = 2$

Subtract from both sides:

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

Simplify the arithmetic:

$2 y = 2 - 3$

Simplify the arithmetic:

$2 y = - 1$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{- 1}{2}$

Simplify the fraction:

$y = \frac{- 1}{2}$

12 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 4 y + 3 = (3 y - 2) - 3 y$

Group like terms:

$- 4 y + 3 = (3 y - 3 y) - 2$

Simplify the arithmetic:

$- 4 y + 3 = - 2$

Subtract from both sides:

$(- 4 y + 3) - 3 = - 2 - 3$

Simplify the arithmetic:

$- 4 y = - 2 - 3$

Simplify the arithmetic:

$- 4 y = - 5$

Divide both sides by :

$\frac{(- 4 y)}{- 4} = \frac{- 5}{- 4}$

Cancel out the negatives:

$\frac{4 y}{4} = \frac{- 5}{- 4}$

Simplify the fraction:

$y = \frac{- 5}{- 4}$

Cancel out the negatives:

$y = \frac{5}{4}$

3. List the solutions

$y = - \frac{1}{2}, \frac{5}{4}$
(2 solution(s))

4. Graph

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