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Solution - Absolute value equations

Exact form:

x = \frac{17}{10}

x=\frac{17}{10}

Mixed number form:

x = 1 \frac{7}{10}

x=1\frac{7}{10}

Decimal form:

x = 1.7

x=1.7

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
$| - x + \frac{2}{5} | = | - x + 3 |$
without the absolute value bars:

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

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:

$\| x \| = \| y \|$	$\| - x + \frac{2}{5} \| = \| - x + 3 \|$
$x = + y$ , $+ x = y$	$(- x + \frac{2}{5}) = (- x + 3)$
$x = - y$ , $- x = y$	$(- x + \frac{2}{5}) = - (- x + 3)$

2. Solve the two equations for x

5 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

The statement is false:

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

The equation is false so it has no solution.

18 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Combine the fractions:

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

Combine the numerators:

$- 2 x + \frac{0}{5} = - 3 - \frac{2}{5}$

Reduce the zero numerator:

$- 2 x + 0 = - 3 - \frac{2}{5}$

Simplify the arithmetic:

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

Convert the integer into a fraction:

$- 2 x = \frac{- 15}{5} + \frac{- 2}{5}$

Combine the fractions:

$- 2 x = \frac{(- 15 - 2)}{5}$

Combine the numerators:

$- 2 x = \frac{- 17}{5}$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{(\frac{- 17}{5})}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{(\frac{- 17}{5})}{- 2}$

Simplify the fraction:

$x = \frac{(\frac{- 17}{5})}{- 2}$

Simplify the arithmetic:

$x = \frac{- 17}{(5 \cdot - 2)}$

$x = \frac{17}{10}$

3. Graph

Each line represents the function of one side of the equation:
$y = | - x + \frac{2}{5} |$
$y = | - x + 3 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved