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Solution - Absolute value equations

Exact form:

y = - \frac{3}{14}, \frac{3}{2}

y=-\frac{3}{14} , \frac{3}{2}

Mixed number form:

y = - \frac{3}{14}, 1 \frac{1}{2}

y=-\frac{3}{14} , 1\frac{1}{2}

Decimal form:

y = - 0.214, 1.5

y=-0.214 , 1.5

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$\frac{1}{2} | 12 y + 6 | - | - 8 y | = 0$

Add $| - 8 y |$ to both sides of the equation:

$\frac{1}{2} | 12 y + 6 | - | - 8 y | + | - 8 y | = | - 8 y |$

Simplify the arithmetic

$\frac{1}{2} | 12 y + 6 | = | - 8 y |$

2. 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
$\frac{1}{2} | 12 y + 6 | = | - 8 y |$
without the absolute value bars:

$\| x \| = \| y \|$	$\frac{1}{2} \| 12 y + 6 \| = \| - 8 y \|$
$x = + y$	$\frac{1}{2} (12 y + 6) = (- 8 y)$
$x = - y$	$\frac{1}{2} (12 y + 6) = (- (- 8 y))$
$+ x = y$	$\frac{1}{2} (12 y + 6) = (- 8 y)$
$- x = y$	$\frac{1}{2} (- (12 y + 6)) = (- 8 y)$

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:

$\| x \| = \| y \|$	$\frac{1}{2} \| 12 y + 6 \| = \| - 8 y \|$
$x = + y$ , $+ x = y$	$\frac{1}{2} (12 y + 6) = (- 8 y)$
$x = - y$ , $- x = y$	$\frac{1}{2} (12 y + 6) = (- (- 8 y))$

3. Solve the two equations for y

13 additional steps

$\frac{1}{2} \cdot (12 y + 6) = (- 8 y)$

Multiply the fraction(s):

$\frac{(1 \cdot (12 y + 6))}{2} = (- 8 y)$

Break up the fraction:

$\frac{12 y}{2} + \frac{6}{2} = (- 8 y)$

Simplify the fraction:

$6 y + \frac{6}{2} = (- 8 y)$

Find the greatest common factor of the numerator and denominator:

$6 y + \frac{(3 \cdot 2)}{(1 \cdot 2)} = (- 8 y)$

Factor out and cancel the greatest common factor:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$14 y + 3 = (- 8 y) + 8 y$

Simplify the arithmetic:

$14 y + 3 = 0$

Subtract from both sides:

$(14 y + 3) - 3 = 0 - 3$

Simplify the arithmetic:

$14 y = 0 - 3$

Simplify the arithmetic:

$14 y = - 3$

Divide both sides by :

$\frac{(14 y)}{14} = \frac{- 3}{14}$

Simplify the fraction:

$y = \frac{- 3}{14}$

16 additional steps

$\frac{1}{2} \cdot (12 y + 6) = (- (- 8 y))$

Multiply the fraction(s):

$\frac{(1 \cdot (12 y + 6))}{2} = (- (- 8 y))$

Break up the fraction:

$\frac{12 y}{2} + \frac{6}{2} = (- (- 8 y))$

Simplify the fraction:

$6 y + \frac{6}{2} = (- (- 8 y))$

Find the greatest common factor of the numerator and denominator:

$6 y + \frac{(3 \cdot 2)}{(1 \cdot 2)} = (- (- 8 y))$

Factor out and cancel the greatest common factor:

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

Resolve the double minus:

$6 y + 3 = 8 y$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 y + 3 = (8 y) - 8 y$

Simplify the arithmetic:

$- 2 y + 3 = 0$

Subtract from both sides:

$(- 2 y + 3) - 3 = 0 - 3$

Simplify the arithmetic:

$- 2 y = 0 - 3$

Simplify the arithmetic:

$- 2 y = - 3$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

4. List the solutions

$y = - \frac{3}{14}, \frac{3}{2}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = \frac{1}{2} | 12 y + 6 |$
$y = | - 8 y |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for y

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for y

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved