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Solution - Absolute value equations

Exact form:

y = \frac{3}{4}, - \frac{3}{2}

y=\frac{3}{4} , -\frac{3}{2}

Mixed number form:

y = \frac{3}{4}, - 1 \frac{1}{2}

y=\frac{3}{4} , -1\frac{1}{2}

Decimal form:

y = 0.75, - 1.5

y=0.75 , -1.5

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

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

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:

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

2. Solve the two equations for y

5 additional steps

$3 y = (- y + 3)$

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 y = 3$

Divide both sides by :

$\frac{(4 y)}{4} = \frac{3}{4}$

Simplify the fraction:

$y = \frac{3}{4}$

6 additional steps

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

Expand the parentheses:

$3 y = y - 3$

Subtract from both sides:

$(3 y) - y = (y - 3) - y$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 y = - 3$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{- 3}{2}$

Simplify the fraction:

$y = \frac{- 3}{2}$

3. List the solutions

$y = \frac{3}{4}, - \frac{3}{2}$
(2 solution(s))

4. Graph

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