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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = 0.25, 0.5

x=0.25 , 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
$| - 3 x + 1 | = | x |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for x

10 additional steps

$(- 3 x + 1) = x$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 4 x + 1 = x - x$

Simplify the arithmetic:

$- 4 x + 1 = 0$

Subtract from both sides:

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

Simplify the arithmetic:

$- 4 x = 0 - 1$

Simplify the arithmetic:

$- 4 x = - 1$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 1}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 1}{- 4}$

Simplify the fraction:

$x = \frac{- 1}{- 4}$

Cancel out the negatives:

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

10 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 x + 1 = - x + x$

Simplify the arithmetic:

$- 2 x + 1 = 0$

Subtract from both sides:

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

Simplify the arithmetic:

$- 2 x = 0 - 1$

Simplify the arithmetic:

$- 2 x = - 1$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{- 1}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{- 1}{- 2}$

Simplify the fraction:

$x = \frac{- 1}{- 2}$

Cancel out the negatives:

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

3. List the solutions

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

4. Graph

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

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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