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Solution - Absolute value equations

Exact form:

x = \frac{7}{12}, \frac{5}{6}

x=\frac{7}{12} , \frac{5}{6}

Decimal form:

x = 0.583, 0.833

x=0.583 , 0.833

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

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

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:

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

2. Solve the two equations for x

16 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$4 x + \frac{- 1}{3} = (- 3 x + 2) + 3 x$

Group like terms:

$4 x + \frac{- 1}{3} = (- 3 x + 3 x) + 2$

Simplify the arithmetic:

$4 x + \frac{- 1}{3} = 2$

Add to both sides:

$(4 x + \frac{- 1}{3}) + \frac{1}{3} = 2 + \frac{1}{3}$

Combine the fractions:

$4 x + \frac{(- 1 + 1)}{3} = 2 + \frac{1}{3}$

Combine the numerators:

$4 x + \frac{0}{3} = 2 + \frac{1}{3}$

Reduce the zero numerator:

$4 x + 0 = 2 + \frac{1}{3}$

Simplify the arithmetic:

$4 x = 2 + \frac{1}{3}$

Convert the integer into a fraction:

$4 x = \frac{6}{3} + \frac{1}{3}$

Combine the fractions:

$4 x = \frac{(6 + 1)}{3}$

Combine the numerators:

$4 x = \frac{7}{3}$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{(\frac{7}{3})}{4}$

Simplify the fraction:

$x = \frac{(\frac{7}{3})}{4}$

Simplify the arithmetic:

$x = \frac{7}{(3 \cdot 4)}$

$x = \frac{7}{12}$

18 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 x + \frac{- 1}{3} = - 2$

Add to both sides:

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

Combine the fractions:

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

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

$- 2 x = - 2 + \frac{1}{3}$

Convert the integer into a fraction:

$- 2 x = \frac{- 6}{3} + \frac{1}{3}$

Combine the fractions:

$- 2 x = \frac{(- 6 + 1)}{3}$

Combine the numerators:

$- 2 x = \frac{- 5}{3}$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{(\frac{- 5}{3})}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{(\frac{- 5}{3})}{- 2}$

Simplify the fraction:

$x = \frac{(\frac{- 5}{3})}{- 2}$

Simplify the arithmetic:

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

$x = \frac{5}{6}$

3. List the solutions

$x = \frac{7}{12}, \frac{5}{6}$
(2 solution(s))

4. Graph

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