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Solution - Absolute value equations

Exact form:

x = - \frac{19}{42}, - \frac{23}{14}

x=-\frac{19}{42} , -\frac{23}{14}

Mixed number form:

x = - \frac{19}{42}, - 1 \frac{9}{14}

x=-\frac{19}{42} , -1\frac{9}{14}

Decimal form:

x = - 0.452, - 1.643

x=-0.452 , -1.643

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
$| - 2 x + \frac{2}{7} | = | 4 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 x + \frac{2}{7} \| = \| 4 x + 3 \|$
$x = + y$	$(- 2 x + \frac{2}{7}) = (4 x + 3)$
$x = - y$	$(- 2 x + \frac{2}{7}) = - (4 x + 3)$
$+ x = y$	$(- 2 x + \frac{2}{7}) = (4 x + 3)$
$- x = y$	$- (- 2 x + \frac{2}{7}) = (4 x + 3)$

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:

$\| x \| = \| y \|$	$\| - 2 x + \frac{2}{7} \| = \| 4 x + 3 \|$
$x = + y$ , $+ x = y$	$(- 2 x + \frac{2}{7}) = (4 x + 3)$
$x = - y$ , $- x = y$	$(- 2 x + \frac{2}{7}) = - (4 x + 3)$

2. Solve the two equations for x

17 additional steps

$(- 2 x + \frac{2}{7}) = (4 x + 3)$

Subtract from both sides:

$(- 2 x + \frac{2}{7}) - 4 x = (4 x + 3) - 4 x$

Group like terms:

$(- 2 x - 4 x) + \frac{2}{7} = (4 x + 3) - 4 x$

Simplify the arithmetic:

$- 6 x + \frac{2}{7} = (4 x + 3) - 4 x$

Group like terms:

$- 6 x + \frac{2}{7} = (4 x - 4 x) + 3$

Simplify the arithmetic:

$- 6 x + \frac{2}{7} = 3$

Subtract from both sides:

$(- 6 x + \frac{2}{7}) - \frac{2}{7} = 3 - \frac{2}{7}$

Combine the fractions:

$- 6 x + \frac{(2 - 2)}{7} = 3 - \frac{2}{7}$

Combine the numerators:

$- 6 x + \frac{0}{7} = 3 - \frac{2}{7}$

Reduce the zero numerator:

$- 6 x + 0 = 3 - \frac{2}{7}$

Simplify the arithmetic:

$- 6 x = 3 - \frac{2}{7}$

Convert the integer into a fraction:

$- 6 x = \frac{21}{7} + \frac{- 2}{7}$

Combine the fractions:

$- 6 x = \frac{(21 - 2)}{7}$

Combine the numerators:

$- 6 x = \frac{19}{7}$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{(\frac{19}{7})}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{(\frac{19}{7})}{- 6}$

Simplify the fraction:

$x = \frac{(\frac{19}{7})}{- 6}$

Simplify the arithmetic:

$x = \frac{19}{(7 \cdot - 6)}$

$x = \frac{- 19}{42}$

17 additional steps

$(- 2 x + \frac{2}{7}) = - (4 x + 3)$

Expand the parentheses:

$(- 2 x + \frac{2}{7}) = - 4 x - 3$

Add to both sides:

$(- 2 x + \frac{2}{7}) + 4 x = (- 4 x - 3) + 4 x$

Group like terms:

$(- 2 x + 4 x) + \frac{2}{7} = (- 4 x - 3) + 4 x$

Simplify the arithmetic:

$2 x + \frac{2}{7} = (- 4 x - 3) + 4 x$

Group like terms:

$2 x + \frac{2}{7} = (- 4 x + 4 x) - 3$

Simplify the arithmetic:

$2 x + \frac{2}{7} = - 3$

Subtract from both sides:

$(2 x + \frac{2}{7}) - \frac{2}{7} = - 3 - \frac{2}{7}$

Combine the fractions:

$2 x + \frac{(2 - 2)}{7} = - 3 - \frac{2}{7}$

Combine the numerators:

$2 x + \frac{0}{7} = - 3 - \frac{2}{7}$

Reduce the zero numerator:

$2 x + 0 = - 3 - \frac{2}{7}$

Simplify the arithmetic:

$2 x = - 3 - \frac{2}{7}$

Convert the integer into a fraction:

$2 x = \frac{- 21}{7} + \frac{- 2}{7}$

Combine the fractions:

$2 x = \frac{(- 21 - 2)}{7}$

Combine the numerators:

$2 x = \frac{- 23}{7}$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{(\frac{- 23}{7})}{2}$

Simplify the fraction:

$x = \frac{(\frac{- 23}{7})}{2}$

Simplify the arithmetic:

$x = \frac{- 23}{(7 \cdot 2)}$

$x = \frac{- 23}{14}$

3. List the solutions

$x = - \frac{19}{42}, - \frac{23}{14}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 x + \frac{2}{7} |$
$y = | 4 x + 3 |$
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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