Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{4}{3}, \frac{2}{9}

x=\frac{4}{3} , \frac{2}{9}

Mixed number form:

x = 1 \frac{1}{3}, \frac{2}{9}

x=1\frac{1}{3} , \frac{2}{9}

Decimal form:

x = 1.333, 0.222

x=1.333 , 0.222

See steps

Step-by-step explanation

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

$| 2 x - 1 | - | x + \frac{1}{3} | = 0$

Add $| x + \frac{1}{3} |$ to both sides of the equation:

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

9 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

$x = \frac{1}{3} + \frac{3}{3}$

Combine the fractions:

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

Combine the numerators:

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

14 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

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

Combine the fractions:

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

Combine the numerators:

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

Divide both sides by :

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

Simplify the fraction:

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

Simplify the arithmetic:

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

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

4. List the solutions

$x = \frac{4}{3}, \frac{2}{9}$
(2 solution(s))

5. Graph

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