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Solution - Absolute value equations

Exact form:

y = \frac{15}{2}, - \frac{35}{6}

y=\frac{15}{2} , -\frac{35}{6}

Mixed number form:

y = 7 \frac{1}{2}, - 5 \frac{5}{6}

y=7\frac{1}{2} , -5\frac{5}{6}

Decimal form:

y = 7.5, - 5.833

y=7.5 , -5.833

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

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

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:

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

2. Solve the two equations for y

20 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

$\frac{- 2}{5} \cdot y + 5 = \frac{0}{5} y + 2$

Reduce the zero numerator:

$\frac{- 2}{5} y + 5 = 0 y + 2$

Simplify the arithmetic:

$\frac{- 2}{5} y + 5 = 2$

Subtract from both sides:

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

Simplify the arithmetic:

$\frac{- 2}{5} y = 2 - 5$

Simplify the arithmetic:

$\frac{- 2}{5} y = - 3$

Multiply both sides by inverse fraction :

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

Move the negative sign from the denominator to the numerator:

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the arithmetic:

$1 y = - 3 \cdot \frac{5}{- 2}$

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

Move the negative sign from the denominator to the numerator:

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

Multiply the fraction(s):

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

Simplify the arithmetic:

$y = \frac{15}{2}$

18 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

$\frac{6}{5} \cdot y + 5 = (\frac{- 4}{5} \cdot y - 2) + \frac{4}{5} y$

Group like terms:

$\frac{6}{5} \cdot y + 5 = (\frac{- 4}{5} \cdot y + \frac{4}{5} y) - 2$

Combine the fractions:

$\frac{6}{5} \cdot y + 5 = \frac{(- 4 + 4)}{5} y - 2$

Combine the numerators:

$\frac{6}{5} \cdot y + 5 = \frac{0}{5} y - 2$

Reduce the zero numerator:

$\frac{6}{5} y + 5 = 0 y - 2$

Simplify the arithmetic:

$\frac{6}{5} y + 5 = - 2$

Subtract from both sides:

$(\frac{6}{5} y + 5) - 5 = - 2 - 5$

Simplify the arithmetic:

$\frac{6}{5} y = - 2 - 5$

Simplify the arithmetic:

$\frac{6}{5} y = - 7$

Multiply both sides by inverse fraction :

$(\frac{6}{5} y) \cdot \frac{5}{6} = - 7 \cdot \frac{5}{6}$

Group like terms:

$(\frac{6}{5} \cdot \frac{5}{6}) y = - 7 \cdot \frac{5}{6}$

Multiply the coefficients:

$\frac{(6 \cdot 5)}{(5 \cdot 6)} y = - 7 \cdot \frac{5}{6}$

Simplify the fraction:

$y = - 7 \cdot \frac{5}{6}$

Multiply the fraction(s):

$y = \frac{(- 7 \cdot 5)}{6}$

Simplify the arithmetic:

$y = \frac{- 35}{6}$

3. List the solutions

$y = \frac{15}{2}, - \frac{35}{6}$
(2 solution(s))

4. Graph

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

Latest Related Drills Solved