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Solution - Absolute value equations

Exact form:

= \frac{3}{10}, \frac{7}{10}

=\frac{3}{10} , \frac{7}{10}

Decimal form:

= 0.3, 0.7

=0.3 , 0.7

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
$| + \frac{1}{5} | = | - x + \frac{1}{2} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| + \frac{1}{5} \| = \| - x + \frac{1}{2} \|$
$x = + y$	$(+ \frac{1}{5}) = (- x + \frac{1}{2})$
$x = - y$	$(+ \frac{1}{5}) = - (- x + \frac{1}{2})$
$+ x = y$	$(+ \frac{1}{5}) = (- x + \frac{1}{2})$
$- x = y$	$- (+ \frac{1}{5}) = (- x + \frac{1}{2})$

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:

$\| x \| = \| y \|$	$\| + \frac{1}{5} \| = \| - x + \frac{1}{2} \|$
$x = + y$ , $+ x = y$	$(+ \frac{1}{5}) = (- x + \frac{1}{2})$
$x = - y$ , $- x = y$	$(+ \frac{1}{5}) = - (- x + \frac{1}{2})$

2. Solve the two equations for

13 additional steps

$(\frac{1}{5}) = (- x + \frac{1}{2})$

Swap sides:

$(- x + \frac{1}{2}) = (\frac{1}{5})$

Subtract from both sides:

$(- x + \frac{1}{2}) - \frac{1}{2} = (\frac{1}{5}) - \frac{1}{2}$

Combine the fractions:

$- x + \frac{(1 - 1)}{2} = (\frac{1}{5}) - \frac{1}{2}$

Combine the numerators:

$- x + \frac{0}{2} = (\frac{1}{5}) - \frac{1}{2}$

Reduce the zero numerator:

$- x + 0 = (\frac{1}{5}) - \frac{1}{2}$

Simplify the arithmetic:

$- x = (\frac{1}{5}) - \frac{1}{2}$

Find the lowest common denominator:

$- x = \frac{(1 \cdot 2)}{(5 \cdot 2)} + \frac{(- 1 \cdot 5)}{(2 \cdot 5)}$

Multiply the denominators:

$- x = \frac{(1 \cdot 2)}{10} + \frac{(- 1 \cdot 5)}{10}$

Multiply the numerators:

$- x = \frac{2}{10} + \frac{- 5}{10}$

Combine the fractions:

$- x = \frac{(2 - 5)}{10}$

Combine the numerators:

$- x = \frac{- 3}{10}$

Multiply both sides by :

$- x \cdot - 1 = (\frac{- 3}{10}) \cdot - 1$

Remove the one(s):

$x = (\frac{- 3}{10}) \cdot - 1$

Remove the one(s):

$x = \frac{3}{10}$

11 additional steps

$(\frac{1}{5}) = - (- x + \frac{1}{2})$

Expand the parentheses:

$(\frac{1}{5}) = x + \frac{- 1}{2}$

Swap sides:

$x + \frac{- 1}{2} = (\frac{1}{5})$

Add to both sides:

$(x + \frac{- 1}{2}) + \frac{1}{2} = (\frac{1}{5}) + \frac{1}{2}$

Combine the fractions:

$x + \frac{(- 1 + 1)}{2} = (\frac{1}{5}) + \frac{1}{2}$

Combine the numerators:

$x + \frac{0}{2} = (\frac{1}{5}) + \frac{1}{2}$

Reduce the zero numerator:

$x + 0 = (\frac{1}{5}) + \frac{1}{2}$

Simplify the arithmetic:

$x = (\frac{1}{5}) + \frac{1}{2}$

Find the lowest common denominator:

$x = \frac{(1 \cdot 2)}{(5 \cdot 2)} + \frac{(1 \cdot 5)}{(2 \cdot 5)}$

Multiply the denominators:

$x = \frac{(1 \cdot 2)}{10} + \frac{(1 \cdot 5)}{10}$

Multiply the numerators:

$x = \frac{2}{10} + \frac{5}{10}$

Combine the fractions:

$x = \frac{(2 + 5)}{10}$

Combine the numerators:

$x = \frac{7}{10}$

3. List the solutions

$= \frac{3}{10}, \frac{7}{10}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | + \frac{1}{5} |$
$y = | - x + \frac{1}{2} |$
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

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

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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