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Solution - Absolute value equations

Exact form:

x = - 5.333, 0.889

x=-5.333 , 0.889

See steps

Step-by-step explanation

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

$| 0.5 x - 2 | - | x + \frac{2}{3} | = 0$

Add $| x + \frac{2}{3} |$ to both sides of the equation:

$| 0.5 x - 2 | - | x + \frac{2}{3} | + | x + \frac{2}{3} | = | x + \frac{2}{3} |$

Simplify the arithmetic

$| 0.5 x - 2 | = | x + \frac{2}{3} |$

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
$| 0.5 x - 2 | = | x + \frac{2}{3} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 0.5 x - 2 \| = \| x + \frac{2}{3} \|$
$x = + y$	$(0.5 x - 2) = (x + \frac{2}{3})$
$x = - y$	$(0.5 x - 2) = (- (x + \frac{2}{3}))$
$+ x = y$	$(0.5 x - 2) = (x + \frac{2}{3})$
$- x = y$	$- (0.5 x - 2) = (x + \frac{2}{3})$

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:

$\| x \| = \| y \|$	$\| 0.5 x - 2 \| = \| x + \frac{2}{3} \|$
$x = + y$ , $+ x = y$	$(0.5 x - 2) = (x + \frac{2}{3})$
$x = - y$ , $- x = y$	$(0.5 x - 2) = (- (x + \frac{2}{3}))$

3. Solve the two equations for x

16 additional steps

$(0.5 x - 2) = (x + \frac{2}{3})$

Subtract from both sides:

$(0.5 x - 2) - x = (x + \frac{2}{3}) - x$

Group like terms:

$(0.5 x - x) - 2 = (x + \frac{2}{3}) - x$

Simplify the arithmetic:

$- 0.5 x - 2 = (x + \frac{2}{3}) - x$

Group like terms:

$- 0.5 x - 2 = (x - x) + \frac{2}{3}$

Simplify the arithmetic:

$- 0.5 x - 2 = \frac{2}{3}$

Add to both sides:

$(- 0.5 x - 2) + 2 = (\frac{2}{3}) + 2$

Simplify the arithmetic:

$- 0.5 x = (\frac{2}{3}) + 2$

Convert the integer into a fraction:

$- 0.5 x = \frac{2}{3} + \frac{6}{3}$

Combine the fractions:

$- 0.5 x = \frac{(2 + 6)}{3}$

Combine the numerators:

$- 0.5 x = \frac{8}{3}$

Divide both sides by :

$\frac{(- 0.5 x)}{- 0.5} = \frac{(\frac{8}{3})}{- 0.5}$

Cancel out the negatives:

$\frac{0.5 x}{0.5} = \frac{(\frac{8}{3})}{- 0.5}$

Simplify the arithmetic:

$x = \frac{(\frac{8}{3})}{- 0.5}$

Simplify the arithmetic:

$x = \frac{8}{(3 \cdot - 0.5)}$

$x = \frac{8}{- 1.5}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 8}{1.5}$

Simplify the arithmetic:

$x = - 5.3333$

15 additional steps

$(0.5 x - 2) = - (x + \frac{2}{3})$

Expand the parentheses:

$(0.5 x - 2) = - x + \frac{- 2}{3}$

Add to both sides:

$(0.5 x - 2) + x = (- x + \frac{- 2}{3}) + x$

Group like terms:

$(0.5 x + x) - 2 = (- x + \frac{- 2}{3}) + x$

Simplify the arithmetic:

$1.5 x - 2 = (- x + \frac{- 2}{3}) + x$

Group like terms:

$1.5 x - 2 = (- x + x) + \frac{- 2}{3}$

Simplify the arithmetic:

$1.5 x - 2 = \frac{- 2}{3}$

Add to both sides:

$(1.5 x - 2) + 2 = (\frac{- 2}{3}) + 2$

Simplify the arithmetic:

$1.5 x = (\frac{- 2}{3}) + 2$

Convert the integer into a fraction:

$1.5 x = \frac{- 2}{3} + \frac{6}{3}$

Combine the fractions:

$1.5 x = \frac{(- 2 + 6)}{3}$

Combine the numerators:

$1.5 x = \frac{4}{3}$

Divide both sides by :

$\frac{(1.5 x)}{1.5} = \frac{(\frac{4}{3})}{1.5}$

Simplify the arithmetic:

$x = \frac{(\frac{4}{3})}{1.5}$

Simplify the arithmetic:

$x = \frac{4}{(3 \cdot 1.5)}$

$x = \frac{4}{4.5}$

$x = 0.8889$

4. List the solutions

$x = - 5.333, 0.889$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 0.5 x - 2 |$
$y = | x + \frac{2}{3} |$
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 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