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Solution - Absolute value equations

Exact form:

x = \frac{11}{3}, - \frac{7}{9}

x=\frac{11}{3} , -\frac{7}{9}

Mixed number form:

x = 3 \frac{2}{3}, - \frac{7}{9}

x=3\frac{2}{3} , -\frac{7}{9}

Decimal form:

x = 3.667, - 0.778

x=3.667 , -0.778

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

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

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:

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

2. Solve the two equations for x

14 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

$- x = \frac{- 2}{3} + \frac{- 9}{3}$

Combine the fractions:

$- x = \frac{(- 2 - 9)}{3}$

Combine the numerators:

$- x = \frac{- 11}{3}$

Multiply both sides by :

$- x \cdot - 1 = (\frac{- 11}{3}) \cdot - 1$

Remove the one(s):

$x = (\frac{- 11}{3}) \cdot - 1$

Remove the one(s):

$x = \frac{11}{3}$

18 additional steps

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

Expand the parentheses:

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

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

$3 x = \frac{2}{3} + \frac{- 9}{3}$

Combine the fractions:

$3 x = \frac{(2 - 9)}{3}$

Combine the numerators:

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

Divide both sides by :

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

Simplify the fraction:

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

Simplify the arithmetic:

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

$x = \frac{- 7}{9}$

3. List the solutions

$x = \frac{11}{3}, - \frac{7}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x + 3 |$
$y = 2 | x - \frac{1}{3} |$
The equation is true where the two lines cross.

How did we do?

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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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