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Solution - Absolute value equations

Exact form:

y = - \frac{9}{16}, - \frac{15}{32}

y=-\frac{9}{16} , -\frac{15}{32}

Decimal form:

y = - 0.562, - 0.469

y=-0.562 , -0.469

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
$| y + \frac{1}{2} | = | \frac{1}{3} y + \frac{1}{8} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| y + \frac{1}{2} \| = \| \frac{1}{3} y + \frac{1}{8} \|$
$x = + y$	$(y + \frac{1}{2}) = (\frac{1}{3} y + \frac{1}{8})$
$x = - y$	$(y + \frac{1}{2}) = - (\frac{1}{3} y + \frac{1}{8})$
$+ x = y$	$(y + \frac{1}{2}) = (\frac{1}{3} y + \frac{1}{8})$
$- x = y$	$- (y + \frac{1}{2}) = (\frac{1}{3} y + \frac{1}{8})$

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:

$\| x \| = \| y \|$	$\| y + \frac{1}{2} \| = \| \frac{1}{3} y + \frac{1}{8} \|$
$x = + y$ , $+ x = y$	$(y + \frac{1}{2}) = (\frac{1}{3} y + \frac{1}{8})$
$x = - y$ , $- x = y$	$(y + \frac{1}{2}) = - (\frac{1}{3} y + \frac{1}{8})$

2. Solve the two equations for y

27 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

$\frac{2}{3} \cdot y + \frac{1}{2} = \frac{0}{3} y + \frac{1}{8}$

Reduce the zero numerator:

$\frac{2}{3} y + \frac{1}{2} = 0 y + \frac{1}{8}$

Simplify the arithmetic:

$\frac{2}{3} y + \frac{1}{2} = \frac{1}{8}$

Subtract from both sides:

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

Combine the fractions:

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

Combine the numerators:

$\frac{2}{3} y + \frac{0}{2} = (\frac{1}{8}) - \frac{1}{2}$

Reduce the zero numerator:

$\frac{2}{3} y + 0 = (\frac{1}{8}) - \frac{1}{2}$

Simplify the arithmetic:

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

Find the lowest common denominator:

$\frac{2}{3} y = \frac{1}{8} + \frac{(- 1 \cdot 4)}{(2 \cdot 4)}$

Multiply the denominators:

$\frac{2}{3} y = \frac{1}{8} + \frac{(- 1 \cdot 4)}{8}$

Multiply the numerators:

$\frac{2}{3} y = \frac{1}{8} + \frac{- 4}{8}$

Combine the fractions:

$\frac{2}{3} y = \frac{(1 - 4)}{8}$

Combine the numerators:

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

Multiply both sides by inverse fraction :

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the fraction:

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

Multiply the fraction(s):

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

Simplify the arithmetic:

$y = \frac{- 9}{(8 \cdot 2)}$

$y = \frac{- 9}{16}$

28 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

$\frac{4}{3} \cdot y + \frac{1}{2} = (\frac{- 1}{3} \cdot y + \frac{- 1}{8}) + \frac{1}{3} y$

Group like terms:

$\frac{4}{3} \cdot y + \frac{1}{2} = (\frac{- 1}{3} \cdot y + \frac{1}{3} y) + \frac{- 1}{8}$

Combine the fractions:

$\frac{4}{3} \cdot y + \frac{1}{2} = \frac{(- 1 + 1)}{3} y + \frac{- 1}{8}$

Combine the numerators:

$\frac{4}{3} \cdot y + \frac{1}{2} = \frac{0}{3} y + \frac{- 1}{8}$

Reduce the zero numerator:

$\frac{4}{3} y + \frac{1}{2} = 0 y + \frac{- 1}{8}$

Simplify the arithmetic:

$\frac{4}{3} y + \frac{1}{2} = \frac{- 1}{8}$

Subtract from both sides:

$(\frac{4}{3} y + \frac{1}{2}) - \frac{1}{2} = (\frac{- 1}{8}) - \frac{1}{2}$

Combine the fractions:

$\frac{4}{3} y + \frac{(1 - 1)}{2} = (\frac{- 1}{8}) - \frac{1}{2}$

Combine the numerators:

$\frac{4}{3} y + \frac{0}{2} = (\frac{- 1}{8}) - \frac{1}{2}$

Reduce the zero numerator:

$\frac{4}{3} y + 0 = (\frac{- 1}{8}) - \frac{1}{2}$

Simplify the arithmetic:

$\frac{4}{3} y = (\frac{- 1}{8}) - \frac{1}{2}$

Find the lowest common denominator:

$\frac{4}{3} y = \frac{- 1}{8} + \frac{(- 1 \cdot 4)}{(2 \cdot 4)}$

Multiply the denominators:

$\frac{4}{3} y = \frac{- 1}{8} + \frac{(- 1 \cdot 4)}{8}$

Multiply the numerators:

$\frac{4}{3} y = \frac{- 1}{8} + \frac{- 4}{8}$

Combine the fractions:

$\frac{4}{3} y = \frac{(- 1 - 4)}{8}$

Combine the numerators:

$\frac{4}{3} y = \frac{- 5}{8}$

Multiply both sides by inverse fraction :

$(\frac{4}{3} y) \cdot \frac{3}{4} = (\frac{- 5}{8}) \cdot \frac{3}{4}$

Group like terms:

$(\frac{4}{3} \cdot \frac{3}{4}) y = (\frac{- 5}{8}) \cdot \frac{3}{4}$

Multiply the coefficients:

$\frac{(4 \cdot 3)}{(3 \cdot 4)} y = (\frac{- 5}{8}) \cdot \frac{3}{4}$

Simplify the fraction:

$y = (\frac{- 5}{8}) \cdot \frac{3}{4}$

Multiply the fraction(s):

$y = \frac{(- 5 \cdot 3)}{(8 \cdot 4)}$

Simplify the arithmetic:

$y = \frac{- 15}{(8 \cdot 4)}$

$y = \frac{- 15}{32}$

3. List the solutions

$y = - \frac{9}{16}, - \frac{15}{32}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | y + \frac{1}{2} |$
$y = | \frac{1}{3} y + \frac{1}{8} |$
The equation is true where the two lines cross.

How did we do?

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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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