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Solution - Absolute value equations

Exact form:

y = \frac{7}{5}, - 7

y=\frac{7}{5} , -7

Mixed number form:

y = 1 \frac{2}{5}, - 7

y=1\frac{2}{5} , -7

Decimal form:

y = 1.4, - 7

y=1.4 , -7

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
$| - 2 y + 7 | = | 3 y |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 y + 7 \| = \| 3 y \|$
$x = + y$	$(- 2 y + 7) = (3 y)$
$x = - y$	$(- 2 y + 7) = - (3 y)$
$+ x = y$	$(- 2 y + 7) = (3 y)$
$- x = y$	$- (- 2 y + 7) = (3 y)$

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:

$\| x \| = \| y \|$	$\| - 2 y + 7 \| = \| 3 y \|$
$x = + y$ , $+ x = y$	$(- 2 y + 7) = (3 y)$
$x = - y$ , $- x = y$	$(- 2 y + 7) = - (3 y)$

2. Solve the two equations for y

10 additional steps

$(- 2 y + 7) = 3 y$

Subtract from both sides:

$(- 2 y + 7) - 3 y = (3 y) - 3 y$

Group like terms:

$(- 2 y - 3 y) + 7 = (3 y) - 3 y$

Simplify the arithmetic:

$- 5 y + 7 = (3 y) - 3 y$

Simplify the arithmetic:

$- 5 y + 7 = 0$

Subtract from both sides:

$(- 5 y + 7) - 7 = 0 - 7$

Simplify the arithmetic:

$- 5 y = 0 - 7$

Simplify the arithmetic:

$- 5 y = - 7$

Divide both sides by :

$\frac{(- 5 y)}{- 5} = \frac{- 7}{- 5}$

Cancel out the negatives:

$\frac{5 y}{5} = \frac{- 7}{- 5}$

Simplify the fraction:

$y = \frac{- 7}{- 5}$

Cancel out the negatives:

$y = \frac{7}{5}$

5 additional steps

$(- 2 y + 7) = - 3 y$

Subtract from both sides:

$(- 2 y + 7) - 7 = (- 3 y) - 7$

Simplify the arithmetic:

$- 2 y = (- 3 y) - 7$

Add to both sides:

$(- 2 y) + 3 y = ((- 3 y) - 7) + 3 y$

Simplify the arithmetic:

$y = ((- 3 y) - 7) + 3 y$

Group like terms:

$y = (- 3 y + 3 y) - 7$

Simplify the arithmetic:

$y = - 7$

3. List the solutions

$y = \frac{7}{5}, - 7$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 y + 7 |$
$y = | 3 y |$
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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