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Solution - Absolute value equations

Exact form:

y = 3, \frac{3}{5}

y=3 , \frac{3}{5}

Decimal form:

y = 3, 0.6

y=3 , 0.6

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
$| 4 y | = 3 | 2 y - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 y \| = 3 \| 2 y - 2 \|$
$x = + y$	$(4 y) = 3 (2 y - 2)$
$x = - y$	$(4 y) = 3 (- (2 y - 2))$
$+ x = y$	$(4 y) = 3 (2 y - 2)$
$- x = y$	$- (4 y) = 3 (2 y - 2)$

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:

$\| x \| = \| y \|$	$\| 4 y \| = 3 \| 2 y - 2 \|$
$x = + y$ , $+ x = y$	$(4 y) = 3 (2 y - 2)$
$x = - y$ , $- x = y$	$(4 y) = 3 (- (2 y - 2))$

2. Solve the two equations for y

12 additional steps

$4 y = 3 \cdot (2 y - 2)$

Expand the parentheses:

$4 y = 3 \cdot 2 y + 3 \cdot - 2$

Multiply the coefficients:

$4 y = 6 y + 3 \cdot - 2$

Simplify the arithmetic:

$4 y = 6 y - 6$

Subtract from both sides:

$(4 y) - 6 y = (6 y - 6) - 6 y$

Simplify the arithmetic:

$- 2 y = (6 y - 6) - 6 y$

Group like terms:

$- 2 y = (6 y - 6 y) - 6$

Simplify the arithmetic:

$- 2 y = - 6$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$y = 3$

11 additional steps

$4 y = 3 \cdot (- (2 y - 2))$

Expand the parentheses:

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

Expand the parentheses:

$4 y = 3 \cdot - 2 y + 3 \cdot 2$

Multiply the coefficients:

$4 y = - 6 y + 3 \cdot 2$

Simplify the arithmetic:

$4 y = - 6 y + 6$

Add to both sides:

$(4 y) + 6 y = (- 6 y + 6) + 6 y$

Simplify the arithmetic:

$10 y = (- 6 y + 6) + 6 y$

Group like terms:

$10 y = (- 6 y + 6 y) + 6$

Simplify the arithmetic:

$10 y = 6$

Divide both sides by :

$\frac{(10 y)}{10} = \frac{6}{10}$

Simplify the fraction:

$y = \frac{6}{10}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$y = 3, \frac{3}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 y |$
$y = 3 | 2 y - 2 |$
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

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