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Solution - Absolute value equations

Exact form:

x = 0, \frac{12}{13}

x=0 , \frac{12}{13}

Decimal form:

x = 0, 0.923

x=0 , 0.923

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
$3 | 2 x - 1 | = | \frac{1}{2} x - 3 |$
without the absolute value bars:

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

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:

$\| x \| = \| y \|$	$3 \| 2 x - 1 \| = \| \frac{1}{2} x - 3 \|$
$x = + y$ , $+ x = y$	$3 (2 x - 1) = (\frac{1}{2} x - 3)$
$x = - y$ , $- x = y$	$3 (2 x - 1) = - (\frac{1}{2} x - 3)$

2. Solve the two equations for x

17 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$6 x + 3 \cdot - 1 = (\frac{1}{2} x - 3)$

Simplify the arithmetic:

$6 x - 3 = (\frac{1}{2} x - 3)$

Subtract from both sides:

$(6 x - 3) - \frac{1}{2} \cdot x = (\frac{1}{2} x - 3) - \frac{1}{2} x$

Group like terms:

$(6 x + \frac{- 1}{2} \cdot x) - 3 = (\frac{1}{2} \cdot x - 3) - \frac{1}{2} x$

Group the coefficients:

$(6 + \frac{- 1}{2}) x - 3 = (\frac{1}{2} \cdot x - 3) - \frac{1}{2} x$

Convert the integer into a fraction:

$(\frac{12}{2} + \frac{- 1}{2}) x - 3 = (\frac{1}{2} \cdot x - 3) - \frac{1}{2} x$

Combine the fractions:

$\frac{(12 - 1)}{2} \cdot x - 3 = (\frac{1}{2} \cdot x - 3) - \frac{1}{2} x$

Combine the numerators:

$\frac{11}{2} \cdot x - 3 = (\frac{1}{2} \cdot x - 3) - \frac{1}{2} x$

Group like terms:

$\frac{11}{2} \cdot x - 3 = (\frac{1}{2} \cdot x + \frac{- 1}{2} x) - 3$

Combine the fractions:

$\frac{11}{2} \cdot x - 3 = \frac{(1 - 1)}{2} x - 3$

Combine the numerators:

$\frac{11}{2} \cdot x - 3 = \frac{0}{2} x - 3$

Reduce the zero numerator:

$\frac{11}{2} x - 3 = 0 x - 3$

Simplify the arithmetic:

$\frac{11}{2} x - 3 = - 3$

Add to both sides:

$(\frac{11}{2} x - 3) + 3 = - 3 + 3$

Simplify the arithmetic:

$\frac{11}{2} x = - 3 + 3$

Simplify the arithmetic:

$\frac{11}{2} x = 0$

Divide both sides by the coefficient:

$x = 0$

23 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$6 x + 3 \cdot - 1 = - (\frac{1}{2} x - 3)$

Simplify the arithmetic:

$6 x - 3 = - (\frac{1}{2} x - 3)$

Expand the parentheses:

$6 x - 3 = \frac{- 1}{2} x + 3$

Add to both sides:

$(6 x - 3) + \frac{1}{2} \cdot x = (\frac{- 1}{2} x + 3) + \frac{1}{2} x$

Group like terms:

$(6 x + \frac{1}{2} \cdot x) - 3 = (\frac{- 1}{2} \cdot x + 3) + \frac{1}{2} x$

Group the coefficients:

$(6 + \frac{1}{2}) x - 3 = (\frac{- 1}{2} \cdot x + 3) + \frac{1}{2} x$

Convert the integer into a fraction:

$(\frac{12}{2} + \frac{1}{2}) x - 3 = (\frac{- 1}{2} \cdot x + 3) + \frac{1}{2} x$

Combine the fractions:

$\frac{(12 + 1)}{2} \cdot x - 3 = (\frac{- 1}{2} \cdot x + 3) + \frac{1}{2} x$

Combine the numerators:

$\frac{13}{2} \cdot x - 3 = (\frac{- 1}{2} \cdot x + 3) + \frac{1}{2} x$

Group like terms:

$\frac{13}{2} \cdot x - 3 = (\frac{- 1}{2} \cdot x + \frac{1}{2} x) + 3$

Combine the fractions:

$\frac{13}{2} \cdot x - 3 = \frac{(- 1 + 1)}{2} x + 3$

Combine the numerators:

$\frac{13}{2} \cdot x - 3 = \frac{0}{2} x + 3$

Reduce the zero numerator:

$\frac{13}{2} x - 3 = 0 x + 3$

Simplify the arithmetic:

$\frac{13}{2} x - 3 = 3$

Add to both sides:

$(\frac{13}{2} x - 3) + 3 = 3 + 3$

Simplify the arithmetic:

$\frac{13}{2} x = 3 + 3$

Simplify the arithmetic:

$\frac{13}{2} x = 6$

Multiply both sides by inverse fraction :

$(\frac{13}{2} x) \cdot \frac{2}{13} = 6 \cdot \frac{2}{13}$

Group like terms:

$(\frac{13}{2} \cdot \frac{2}{13}) x = 6 \cdot \frac{2}{13}$

Multiply the coefficients:

$\frac{(13 \cdot 2)}{(2 \cdot 13)} x = 6 \cdot \frac{2}{13}$

Simplify the fraction:

$x = 6 \cdot \frac{2}{13}$

Multiply the fraction(s):

$x = \frac{(6 \cdot 2)}{13}$

Simplify the arithmetic:

$x = \frac{12}{13}$

3. List the solutions

$x = 0, \frac{12}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | 2 x - 1 |$
$y = | \frac{1}{2} x - 3 |$
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 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