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Solution - Absolute value equations

Exact form:

x = \frac{1}{8}, \frac{1}{12}

x=\frac{1}{8} , \frac{1}{12}

Decimal form:

x = 0.125, 0.083

x=0.125 , 0.083

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 5 x - \frac{1}{2} | - | x | = 0$

Add $| x |$ to both sides of the equation:

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

Simplify the arithmetic

$| 5 x - \frac{1}{2} | = | x |$

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

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

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:

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

3. Solve the two equations for x

13 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

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

Add to both sides:

$(4 x + \frac{- 1}{2}) + \frac{1}{2} = 0 + \frac{1}{2}$

Combine the fractions:

$4 x + \frac{(- 1 + 1)}{2} = 0 + \frac{1}{2}$

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

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

Divide both sides by :

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

Simplify the fraction:

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

Simplify the arithmetic:

$x = \frac{1}{(2 \cdot 4)}$

$x = \frac{1}{8}$

13 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$6 x + \frac{- 1}{2} = 0$

Add to both sides:

$(6 x + \frac{- 1}{2}) + \frac{1}{2} = 0 + \frac{1}{2}$

Combine the fractions:

$6 x + \frac{(- 1 + 1)}{2} = 0 + \frac{1}{2}$

Combine the numerators:

$6 x + \frac{0}{2} = 0 + \frac{1}{2}$

Reduce the zero numerator:

$6 x + 0 = 0 + \frac{1}{2}$

Simplify the arithmetic:

$6 x = 0 + \frac{1}{2}$

Simplify the arithmetic:

$6 x = \frac{1}{2}$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the arithmetic:

$x = \frac{1}{(2 \cdot 6)}$

$x = \frac{1}{12}$

4. List the solutions

$x = \frac{1}{8}, \frac{1}{12}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 5 x - \frac{1}{2} |$
$y = | x |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved