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Solution - Absolute value equations

Exact form:

y = \frac{8}{3}, - 8

y=\frac{8}{3} , -8

Mixed number form:

y = 2 \frac{2}{3}, - 8

y=2\frac{2}{3} , -8

Decimal form:

y = 2.667, - 8

y=2.667 , -8

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
$| 3 y - 8 | = | - 3 y + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 y - 8 \| = \| - 3 y + 8 \|$
$x = + y$	$(3 y - 8) = (- 3 y + 8)$
$x = - y$	$(3 y - 8) = - (- 3 y + 8)$
$+ x = y$	$(3 y - 8) = (- 3 y + 8)$
$- x = y$	$- (3 y - 8) = (- 3 y + 8)$

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:

$\| x \| = \| y \|$	$\| 3 y - 8 \| = \| - 3 y + 8 \|$
$x = + y$ , $+ x = y$	$(3 y - 8) = (- 3 y + 8)$
$x = - y$ , $- x = y$	$(3 y - 8) = - (- 3 y + 8)$

2. Solve the two equations for y

11 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$6 y - 8 = (- 3 y + 8) + 3 y$

Group like terms:

$6 y - 8 = (- 3 y + 3 y) + 8$

Simplify the arithmetic:

$6 y - 8 = 8$

Add to both sides:

$(6 y - 8) + 8 = 8 + 8$

Simplify the arithmetic:

$6 y = 8 + 8$

Simplify the arithmetic:

$6 y = 16$

Divide both sides by :

$\frac{(6 y)}{6} = \frac{16}{6}$

Simplify the fraction:

$y = \frac{16}{6}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(8 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$y = \frac{8}{3}$

5 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 8 = - 8$

3. List the solutions

$y = \frac{8}{3}, - 8$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 y - 8 |$
$y = | - 3 y + 8 |$
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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