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Solution - Absolute value equations

Exact form:

y = \frac{1}{2}, \frac{1}{4}

y=\frac{1}{2} , \frac{1}{4}

Decimal form:

y = 0.5, 0.25

y=0.5 , 0.25

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
$| 3 y - 1 | = | y |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 y - 1 \| = \| y \|$
$x = + y$	$(3 y - 1) = (y)$
$x = - y$	$(3 y - 1) = - (y)$
$+ x = y$	$(3 y - 1) = (y)$
$- x = y$	$- (3 y - 1) = (y)$

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:

$\| x \| = \| y \|$	$\| 3 y - 1 \| = \| y \|$
$x = + y$ , $+ x = y$	$(3 y - 1) = (y)$
$x = - y$ , $- x = y$	$(3 y - 1) = - (y)$

2. Solve the two equations for y

8 additional steps

$(3 y - 1) = y$

Subtract from both sides:

$(3 y - 1) - y = y - y$

Group like terms:

$(3 y - y) - 1 = y - y$

Simplify the arithmetic:

$2 y - 1 = y - y$

Simplify the arithmetic:

$2 y - 1 = 0$

Add to both sides:

$(2 y - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$2 y = 0 + 1$

Simplify the arithmetic:

$2 y = 1$

Divide both sides by :

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

Simplify the fraction:

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

8 additional steps

$(3 y - 1) = - y$

Add to both sides:

$(3 y - 1) + y = - y + y$

Group like terms:

$(3 y + y) - 1 = - y + y$

Simplify the arithmetic:

$4 y - 1 = - y + y$

Simplify the arithmetic:

$4 y - 1 = 0$

Add to both sides:

$(4 y - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$4 y = 0 + 1$

Simplify the arithmetic:

$4 y = 1$

Divide both sides by :

$\frac{(4 y)}{4} = \frac{1}{4}$

Simplify the fraction:

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

3. List the solutions

$y = \frac{1}{2}, \frac{1}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 y - 1 |$
$y = | y |$
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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