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Solution - Absolute value equations

Exact form:

x = \frac{4}{3}, 4

x=\frac{4}{3} , 4

Mixed number form:

x = 1 \frac{1}{3}, 4

x=1\frac{1}{3} , 4

Decimal form:

x = 1.333, 4

x=1.333 , 4

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

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

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:

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

2. Solve the two equations for x

21 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Combine the fractions:

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

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

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

Multiply both sides by inverse fraction :

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

Move the negative sign from the denominator to the numerator:

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

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

Move the negative sign from the denominator to the numerator:

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

Multiply the fraction(s):

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

Simplify the arithmetic:

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

21 additional steps

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

Group like terms:

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

Multiply the coefficients:

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

Combine like terms:

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Combine the fractions:

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

Combine the numerators:

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

Reduce the zero numerator:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

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

Multiply both sides by inverse fraction :

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the arithmetic:

$1 x = - 2 \cdot \frac{2}{- 1}$

$x = - 2 \cdot \frac{2}{- 1}$

Simplify the arithmetic:

$x = 4$

3. List the solutions

$x = \frac{4}{3}, 4$
(2 solution(s))

4. Graph

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

Latest Related Drills Solved