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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

y = - 1 \frac{1}{2}, \frac{1}{2}

y=-1\frac{1}{2} , \frac{1}{2}

Decimal form:

y = - 1.5, 0.5

y=-1.5 , 0.5

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

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

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:

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

2. Solve the two equations for y

10 additional steps

$(2 y - 3) = 4 y$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$- 2 y - 3 = 0$

Add to both sides:

$(- 2 y - 3) + 3 = 0 + 3$

Simplify the arithmetic:

$- 2 y = 0 + 3$

Simplify the arithmetic:

$- 2 y = 3$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

9 additional steps

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

Add to both sides:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

$6 y = ((- 4 y) + 3) + 4 y$

Group like terms:

$6 y = (- 4 y + 4 y) + 3$

Simplify the arithmetic:

$6 y = 3$

Divide both sides by :

$\frac{(6 y)}{6} = \frac{3}{6}$

Simplify the fraction:

$y = \frac{3}{6}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$y = - \frac{3}{2}, \frac{1}{2}$
(2 solution(s))

4. Graph

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