Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{10}{7}, - \frac{2}{3}

x=\frac{10}{7} , -\frac{2}{3}

Mixed number form:

x = 1 \frac{3}{7}, - \frac{2}{3}

x=1\frac{3}{7} , -\frac{2}{3}

Decimal form:

x = 1.429, - 0.667

x=1.429 , -0.667

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
$\frac{1}{2} | x + 8 | = | 4 x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\frac{1}{2} \| x + 8 \| = \| 4 x - 1 \|$
$x = + y$	$\frac{1}{2} (x + 8) = (4 x - 1)$
$x = - y$	$\frac{1}{2} (x + 8) = - (4 x - 1)$
$+ x = y$	$\frac{1}{2} (x + 8) = (4 x - 1)$
$- x = y$	$\frac{1}{2} (- (x + 8)) = (4 x - 1)$

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:

$\| x \| = \| y \|$	$\frac{1}{2} \| x + 8 \| = \| 4 x - 1 \|$
$x = + y$ , $+ x = y$	$\frac{1}{2} (x + 8) = (4 x - 1)$
$x = - y$ , $- x = y$	$\frac{1}{2} (x + 8) = - (4 x - 1)$

2. Solve the two equations for x

23 additional steps

$\frac{1}{2} \cdot (x + 8) = (4 x - 1)$

Multiply the fraction(s):

$\frac{(1 \cdot (x + 8))}{2} = (4 x - 1)$

Break up the fraction:

$\frac{x}{2} + \frac{8}{2} = (4 x - 1)$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

Subtract from both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

$(\frac{1}{2} + \frac{- 8}{2}) x + 4 = (4 x - 1) - 4 x$

Combine the fractions:

$\frac{(1 - 8)}{2} x + 4 = (4 x - 1) - 4 x$

Combine the numerators:

$\frac{- 7}{2} x + 4 = (4 x - 1) - 4 x$

Group like terms:

$\frac{- 7}{2} x + 4 = (4 x - 4 x) - 1$

Simplify the arithmetic:

$\frac{- 7}{2} x + 4 = - 1$

Subtract from both sides:

$(\frac{- 7}{2} x + 4) - 4 = - 1 - 4$

Simplify the arithmetic:

$\frac{- 7}{2} x = - 1 - 4$

Simplify the arithmetic:

$\frac{- 7}{2} x = - 5$

Multiply both sides by inverse fraction :

$(\frac{- 7}{2} x) \cdot \frac{2}{- 7} = - 5 \cdot \frac{2}{- 7}$

Move the negative sign from the denominator to the numerator:

$\frac{- 7}{2} x \cdot \frac{- 2}{7} = - 5 \cdot \frac{2}{- 7}$

Group like terms:

$(\frac{- 7}{2} \cdot \frac{- 2}{7}) x = - 5 \cdot \frac{2}{- 7}$

Multiply the coefficients:

$\frac{(- 7 \cdot - 2)}{(2 \cdot 7)} x = - 5 \cdot \frac{2}{- 7}$

Simplify the arithmetic:

$1 x = - 5 \cdot \frac{2}{- 7}$

$x = - 5 \cdot \frac{2}{- 7}$

Move the negative sign from the denominator to the numerator:

$x = - 5 \cdot \frac{- 2}{7}$

Multiply the fraction(s):

$x = \frac{(- 5 \cdot - 2)}{7}$

Simplify the arithmetic:

$x = \frac{10}{7}$

21 additional steps

$\frac{1}{2} \cdot (x + 8) = - (4 x - 1)$

Multiply the fraction(s):

$\frac{(1 \cdot (x + 8))}{2} = - (4 x - 1)$

Break up the fraction:

$\frac{x}{2} + \frac{8}{2} = - (4 x - 1)$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Group the coefficients:

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

Convert the integer into a fraction:

$(\frac{1}{2} + \frac{8}{2}) x + 4 = (- 4 x + 1) + 4 x$

Combine the fractions:

$\frac{(1 + 8)}{2} x + 4 = (- 4 x + 1) + 4 x$

Combine the numerators:

$\frac{9}{2} x + 4 = (- 4 x + 1) + 4 x$

Group like terms:

$\frac{9}{2} x + 4 = (- 4 x + 4 x) + 1$

Simplify the arithmetic:

$\frac{9}{2} x + 4 = 1$

Subtract from both sides:

$(\frac{9}{2} x + 4) - 4 = 1 - 4$

Simplify the arithmetic:

$\frac{9}{2} x = 1 - 4$

Simplify the arithmetic:

$\frac{9}{2} x = - 3$

Multiply both sides by inverse fraction :

$(\frac{9}{2} x) \cdot \frac{2}{9} = - 3 \cdot \frac{2}{9}$

Group like terms:

$(\frac{9}{2} \cdot \frac{2}{9}) x = - 3 \cdot \frac{2}{9}$

Multiply the coefficients:

$\frac{(9 \cdot 2)}{(2 \cdot 9)} x = - 3 \cdot \frac{2}{9}$

Simplify the fraction:

$x = - 3 \cdot \frac{2}{9}$

Multiply the fraction(s):

$x = \frac{(- 3 \cdot 2)}{9}$

Simplify the arithmetic:

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

3. List the solutions

$x = \frac{10}{7}, - \frac{2}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = \frac{1}{2} | x + 8 |$
$y = | 4 x - 1 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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