Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = 2, \frac{6}{7}

x=2 , \frac{6}{7}

Decimal form:

x = 2, 0.857

x=2 , 0.857

See steps

Step-by-step explanation

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

$| 7 x - 2 | - 2 | 7 x - 8 | = 0$

Add $2 | 7 x - 8 |$ to both sides of the equation:

$| 7 x - 2 | - 2 | 7 x - 8 | + 2 | 7 x - 8 | = 2 | 7 x - 8 |$

Simplify the arithmetic

$| 7 x - 2 | = 2 | 7 x - 8 |$

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
$| 7 x - 2 | = 2 | 7 x - 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 x - 2 \| = 2 \| 7 x - 8 \|$
$x = + y$	$(7 x - 2) = 2 (7 x - 8)$
$x = - y$	$(7 x - 2) = 2 (- (7 x - 8))$
$+ x = y$	$(7 x - 2) = 2 (7 x - 8)$
$- x = y$	$- (7 x - 2) = 2 (7 x - 8)$

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:

$\| x \| = \| y \|$	$\| 7 x - 2 \| = 2 \| 7 x - 8 \|$
$x = + y$ , $+ x = y$	$(7 x - 2) = 2 (7 x - 8)$
$x = - y$ , $- x = y$	$(7 x - 2) = 2 (- (7 x - 8))$

3. Solve the two equations for x

16 additional steps

$(7 x - 2) = 2 \cdot (7 x - 8)$

Expand the parentheses:

$(7 x - 2) = 2 \cdot 7 x + 2 \cdot - 8$

Multiply the coefficients:

$(7 x - 2) = 14 x + 2 \cdot - 8$

Simplify the arithmetic:

$(7 x - 2) = 14 x - 16$

Subtract from both sides:

$(7 x - 2) - 14 x = (14 x - 16) - 14 x$

Group like terms:

$(7 x - 14 x) - 2 = (14 x - 16) - 14 x$

Simplify the arithmetic:

$- 7 x - 2 = (14 x - 16) - 14 x$

Group like terms:

$- 7 x - 2 = (14 x - 14 x) - 16$

Simplify the arithmetic:

$- 7 x - 2 = - 16$

Add to both sides:

$(- 7 x - 2) + 2 = - 16 + 2$

Simplify the arithmetic:

$- 7 x = - 16 + 2$

Simplify the arithmetic:

$- 7 x = - 14$

Divide both sides by :

$\frac{(- 7 x)}{- 7} = \frac{- 14}{- 7}$

Cancel out the negatives:

$\frac{7 x}{7} = \frac{- 14}{- 7}$

Simplify the fraction:

$x = \frac{- 14}{- 7}$

Cancel out the negatives:

$x = \frac{14}{7}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 7)}{(1 \cdot 7)}$

Factor out and cancel the greatest common factor:

$x = 2$

15 additional steps

$(7 x - 2) = 2 \cdot (- (7 x - 8))$

Expand the parentheses:

$(7 x - 2) = 2 \cdot (- 7 x + 8)$

Expand the parentheses:

$(7 x - 2) = 2 \cdot - 7 x + 2 \cdot 8$

Multiply the coefficients:

$(7 x - 2) = - 14 x + 2 \cdot 8$

Simplify the arithmetic:

$(7 x - 2) = - 14 x + 16$

Add to both sides:

$(7 x - 2) + 14 x = (- 14 x + 16) + 14 x$

Group like terms:

$(7 x + 14 x) - 2 = (- 14 x + 16) + 14 x$

Simplify the arithmetic:

$21 x - 2 = (- 14 x + 16) + 14 x$

Group like terms:

$21 x - 2 = (- 14 x + 14 x) + 16$

Simplify the arithmetic:

$21 x - 2 = 16$

Add to both sides:

$(21 x - 2) + 2 = 16 + 2$

Simplify the arithmetic:

$21 x = 16 + 2$

Simplify the arithmetic:

$21 x = 18$

Divide both sides by :

$\frac{(21 x)}{21} = \frac{18}{21}$

Simplify the fraction:

$x = \frac{18}{21}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(6 \cdot 3)}{(7 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{6}{7}$

4. List the solutions

$x = 2, \frac{6}{7}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 7 x - 2 |$
$y = 2 | 7 x - 8 |$
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