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Solution - Absolute value equations

Exact form:

x = \frac{19}{6}, \frac{21}{10}

x=\frac{19}{6} , \frac{21}{10}

Mixed number form:

x = 3 \frac{1}{6}, 2 \frac{1}{10}

x=3\frac{1}{6} , 2\frac{1}{10}

Decimal form:

x = 3.167, 2.1

x=3.167 , 2.1

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
$| 4 x - 10 | = | x - \frac{1}{2} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 10 \| = \| x - \frac{1}{2} \|$
$x = + y$	$(4 x - 10) = (x - \frac{1}{2})$
$x = - y$	$(4 x - 10) = - (x - \frac{1}{2})$
$+ x = y$	$(4 x - 10) = (x - \frac{1}{2})$
$- x = y$	$- (4 x - 10) = (x - \frac{1}{2})$

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:

$\| x \| = \| y \|$	$\| 4 x - 10 \| = \| x - \frac{1}{2} \|$
$x = + y$ , $+ x = y$	$(4 x - 10) = (x - \frac{1}{2})$
$x = - y$ , $- x = y$	$(4 x - 10) = - (x - \frac{1}{2})$

2. Solve the two equations for x

13 additional steps

$(4 x - 10) = (x + \frac{- 1}{2})$

Subtract from both sides:

$(4 x - 10) - x = (x + \frac{- 1}{2}) - x$

Group like terms:

$(4 x - x) - 10 = (x + \frac{- 1}{2}) - x$

Simplify the arithmetic:

$3 x - 10 = (x + \frac{- 1}{2}) - x$

Group like terms:

$3 x - 10 = (x - x) + \frac{- 1}{2}$

Simplify the arithmetic:

$3 x - 10 = \frac{- 1}{2}$

Add to both sides:

$(3 x - 10) + 10 = (\frac{- 1}{2}) + 10$

Simplify the arithmetic:

$3 x = (\frac{- 1}{2}) + 10$

Convert the integer into a fraction:

$3 x = \frac{- 1}{2} + \frac{20}{2}$

Combine the fractions:

$3 x = \frac{(- 1 + 20)}{2}$

Combine the numerators:

$3 x = \frac{19}{2}$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{(\frac{19}{2})}{3}$

Simplify the fraction:

$x = \frac{(\frac{19}{2})}{3}$

Simplify the arithmetic:

$x = \frac{19}{(2 \cdot 3)}$

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

14 additional steps

$(4 x - 10) = - (x + \frac{- 1}{2})$

Expand the parentheses:

$(4 x - 10) = - x + \frac{1}{2}$

Add to both sides:

$(4 x - 10) + x = (- x + \frac{1}{2}) + x$

Group like terms:

$(4 x + x) - 10 = (- x + \frac{1}{2}) + x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$5 x - 10 = \frac{1}{2}$

Add to both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

$5 x = \frac{1}{2} + \frac{20}{2}$

Combine the fractions:

$5 x = \frac{(1 + 20)}{2}$

Combine the numerators:

$5 x = \frac{21}{2}$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{(\frac{21}{2})}{5}$

Simplify the fraction:

$x = \frac{(\frac{21}{2})}{5}$

Simplify the arithmetic:

$x = \frac{21}{(2 \cdot 5)}$

$x = \frac{21}{10}$

3. List the solutions

$x = \frac{19}{6}, \frac{21}{10}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 10 |$
$y = | x - \frac{1}{2} |$
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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