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Solution - Absolute value equations

Exact form:

y = - 60, \frac{60}{19}

y=-60 , \frac{60}{19}

Mixed number form:

y = - 60, 3 \frac{3}{19}

y=-60 , 3\frac{3}{19}

Decimal form:

y = - 60, 3.158

y=-60 , 3.158

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
$| \frac{3}{5} y - 4 | = | \frac{2}{3} y |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{3}{5} y - 4 \| = \| \frac{2}{3} y \|$
$x = + y$	$(\frac{3}{5} y - 4) = (\frac{2}{3} y)$
$x = - y$	$(\frac{3}{5} y - 4) = - (\frac{2}{3} y)$
$+ x = y$	$(\frac{3}{5} y - 4) = (\frac{2}{3} y)$
$- x = y$	$- (\frac{3}{5} y - 4) = (\frac{2}{3} y)$

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:

$\| x \| = \| y \|$	$\| \frac{3}{5} y - 4 \| = \| \frac{2}{3} y \|$
$x = + y$ , $+ x = y$	$(\frac{3}{5} y - 4) = (\frac{2}{3} y)$
$x = - y$ , $- x = y$	$(\frac{3}{5} y - 4) = - (\frac{2}{3} y)$

2. Solve the two equations for y

20 additional steps

$(\frac{3}{5} \cdot y - 4) = \frac{2}{3} y$

Subtract from both sides:

$(\frac{3}{5} y - 4) - \frac{2}{3} \cdot y = (\frac{2}{3} y) - \frac{2}{3} y$

Group like terms:

$(\frac{3}{5} \cdot y + \frac{- 2}{3} \cdot y) - 4 = (\frac{2}{3} \cdot y) - \frac{2}{3} y$

Group the coefficients:

$(\frac{3}{5} + \frac{- 2}{3}) y - 4 = (\frac{2}{3} \cdot y) - \frac{2}{3} y$

Find the lowest common denominator:

$(\frac{(3 \cdot 3)}{(5 \cdot 3)} + \frac{(- 2 \cdot 5)}{(3 \cdot 5)}) y - 4 = (\frac{2}{3} \cdot y) - \frac{2}{3} y$

Multiply the denominators:

$(\frac{(3 \cdot 3)}{15} + \frac{(- 2 \cdot 5)}{15}) y - 4 = (\frac{2}{3} \cdot y) - \frac{2}{3} y$

Multiply the numerators:

$(\frac{9}{15} + \frac{- 10}{15}) y - 4 = (\frac{2}{3} \cdot y) - \frac{2}{3} y$

Combine the fractions:

$\frac{(9 - 10)}{15} \cdot y - 4 = (\frac{2}{3} \cdot y) - \frac{2}{3} y$

Combine the numerators:

$\frac{- 1}{15} \cdot y - 4 = (\frac{2}{3} \cdot y) - \frac{2}{3} y$

Combine the fractions:

$\frac{- 1}{15} \cdot y - 4 = \frac{(2 - 2)}{3} y$

Combine the numerators:

$\frac{- 1}{15} \cdot y - 4 = \frac{0}{3} y$

Reduce the zero numerator:

$\frac{- 1}{15} y - 4 = 0 y$

Simplify the arithmetic:

$\frac{- 1}{15} y - 4 = 0$

Add to both sides:

$(\frac{- 1}{15} y - 4) + 4 = 0 + 4$

Simplify the arithmetic:

$\frac{- 1}{15} y = 0 + 4$

Simplify the arithmetic:

$\frac{- 1}{15} y = 4$

Multiply both sides by inverse fraction :

$(\frac{- 1}{15} y) \cdot \frac{15}{- 1} = 4 \cdot \frac{15}{- 1}$

Group like terms:

$(\frac{- 1}{15} \cdot - 15) y = 4 \cdot \frac{15}{- 1}$

Multiply the coefficients:

$\frac{(- 1 \cdot - 15)}{15} y = 4 \cdot \frac{15}{- 1}$

Simplify the arithmetic:

$1 y = 4 \cdot \frac{15}{- 1}$

$y = 4 \cdot \frac{15}{- 1}$

Simplify the arithmetic:

$y = - 60$

19 additional steps

$(\frac{3}{5} \cdot y - 4) = \frac{- 2}{3} y$

Add to both sides:

$(\frac{3}{5} y - 4) + 4 = (\frac{- 2}{3} y) + 4$

Simplify the arithmetic:

$\frac{3}{5} \cdot y = (\frac{- 2}{3} y) + 4$

Add to both sides:

$(\frac{3}{5} y) + \frac{2}{3} \cdot y = (\frac{- 2}{3} y + 4) + \frac{2}{3} y$

Group the coefficients:

$(\frac{3}{5} + \frac{2}{3}) y = (\frac{- 2}{3} \cdot y + 4) + \frac{2}{3} y$

Find the lowest common denominator:

$(\frac{(3 \cdot 3)}{(5 \cdot 3)} + \frac{(2 \cdot 5)}{(3 \cdot 5)}) y = (\frac{- 2}{3} \cdot y + 4) + \frac{2}{3} y$

Multiply the denominators:

$(\frac{(3 \cdot 3)}{15} + \frac{(2 \cdot 5)}{15}) y = (\frac{- 2}{3} \cdot y + 4) + \frac{2}{3} y$

Multiply the numerators:

$(\frac{9}{15} + \frac{10}{15}) y = (\frac{- 2}{3} \cdot y + 4) + \frac{2}{3} y$

Combine the fractions:

$\frac{(9 + 10)}{15} \cdot y = (\frac{- 2}{3} \cdot y + 4) + \frac{2}{3} y$

Combine the numerators:

$\frac{19}{15} \cdot y = (\frac{- 2}{3} \cdot y + 4) + \frac{2}{3} y$

Group like terms:

$\frac{19}{15} \cdot y = (\frac{- 2}{3} \cdot y + \frac{2}{3} y) + 4$

Combine the fractions:

$\frac{19}{15} \cdot y = \frac{(- 2 + 2)}{3} y + 4$

Combine the numerators:

$\frac{19}{15} \cdot y = \frac{0}{3} y + 4$

Reduce the zero numerator:

$\frac{19}{15} y = 0 y + 4$

Simplify the arithmetic:

$\frac{19}{15} y = 4$

Multiply both sides by inverse fraction :

$(\frac{19}{15} y) \cdot \frac{15}{19} = 4 \cdot \frac{15}{19}$

Group like terms:

$(\frac{19}{15} \cdot \frac{15}{19}) y = 4 \cdot \frac{15}{19}$

Multiply the coefficients:

$\frac{(19 \cdot 15)}{(15 \cdot 19)} y = 4 \cdot \frac{15}{19}$

Simplify the fraction:

$y = 4 \cdot \frac{15}{19}$

Multiply the fraction(s):

$y = \frac{(4 \cdot 15)}{19}$

Simplify the arithmetic:

$y = \frac{60}{19}$

3. List the solutions

$y = - 60, \frac{60}{19}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{3}{5} y - 4 |$
$y = | \frac{2}{3} y |$
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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