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Solution - Absolute value equations

Exact form:

y = - 3, \frac{3}{7}

y=-3 , \frac{3}{7}

Decimal form:

y = - 3, 0.429

y=-3 , 0.429

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

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

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:

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

2. Solve the two equations for y

17 additional steps

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

Multiply the fraction(s):

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

Break up the fraction:

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

Subtract from both sides:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Group like terms:

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

Combine the fractions:

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

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

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

Multiply both sides by inverse fraction :

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the fraction:

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

Multiply the fraction(s):

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

Simplify the arithmetic:

$y = - 3$

18 additional steps

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

Multiply the fraction(s):

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

Expand the parentheses:

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

Break up the fraction:

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

Add to both sides:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

$\frac{7}{2} \cdot y = (\frac{- 3 y}{2} + \frac{3}{2}) + \frac{3}{2} y$

Group like terms:

$\frac{7}{2} \cdot y = (\frac{- 3 y}{2} + \frac{3}{2} y) + \frac{3}{2}$

Combine the fractions:

$\frac{7}{2} \cdot y = \frac{(- 3 + 3)}{2} y + \frac{3}{2}$

Combine the numerators:

$\frac{7}{2} \cdot y = \frac{0}{2} y + \frac{3}{2}$

Reduce the zero numerator:

$\frac{7}{2} y = 0 y + \frac{3}{2}$

Simplify the arithmetic:

$\frac{7}{2} y = \frac{3}{2}$

Multiply both sides by inverse fraction :

$(\frac{7}{2} y) \cdot \frac{2}{7} = (\frac{3}{2}) \cdot \frac{2}{7}$

Group like terms:

$(\frac{7}{2} \cdot \frac{2}{7}) y = (\frac{3}{2}) \cdot \frac{2}{7}$

Multiply the coefficients:

$\frac{(7 \cdot 2)}{(2 \cdot 7)} y = (\frac{3}{2}) \cdot \frac{2}{7}$

Simplify the fraction:

$y = (\frac{3}{2}) \cdot \frac{2}{7}$

Multiply the fraction(s):

$y = \frac{(3 \cdot 2)}{(2 \cdot 7)}$

Simplify the arithmetic:

$y = \frac{3}{7}$

3. List the solutions

$y = - 3, \frac{3}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 y |$
$y = \frac{1}{2} | 3 y - 3 |$
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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