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Solution - Absolute value equations

Exact form:

x = \frac{7}{5}, \frac{11}{7}

x=\frac{7}{5} , \frac{11}{7}

Mixed number form:

x = 1 \frac{2}{5}, 1 \frac{4}{7}

x=1\frac{2}{5} , 1\frac{4}{7}

Decimal form:

x = 1.4, 1.571

x=1.4 , 1.571

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
$2 | x - 2 | = 6 | 2 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| x - 2 \| = 6 \| 2 x - 3 \|$
$x = + y$	$2 (x - 2) = 6 (2 x - 3)$
$x = - y$	$2 (x - 2) = 6 (- (2 x - 3))$
$+ x = y$	$2 (x - 2) = 6 (2 x - 3)$
$- x = y$	$2 (- (x - 2)) = 6 (2 x - 3)$

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:

$\| x \| = \| y \|$	$2 \| x - 2 \| = 6 \| 2 x - 3 \|$
$x = + y$ , $+ x = y$	$2 (x - 2) = 6 (2 x - 3)$
$x = - y$ , $- x = y$	$2 (x - 2) = 6 (- (2 x - 3))$

2. Solve the two equations for x

18 additional steps

$2 \cdot (x - 2) = 6 \cdot (2 x - 3)$

Expand the parentheses:

$2 x + 2 \cdot - 2 = 6 \cdot (2 x - 3)$

Simplify the arithmetic:

$2 x - 4 = 6 \cdot (2 x - 3)$

Expand the parentheses:

$2 x - 4 = 6 \cdot 2 x + 6 \cdot - 3$

Multiply the coefficients:

$2 x - 4 = 12 x + 6 \cdot - 3$

Simplify the arithmetic:

$2 x - 4 = 12 x - 18$

Subtract from both sides:

$(2 x - 4) - 12 x = (12 x - 18) - 12 x$

Group like terms:

$(2 x - 12 x) - 4 = (12 x - 18) - 12 x$

Simplify the arithmetic:

$- 10 x - 4 = (12 x - 18) - 12 x$

Group like terms:

$- 10 x - 4 = (12 x - 12 x) - 18$

Simplify the arithmetic:

$- 10 x - 4 = - 18$

Add to both sides:

$(- 10 x - 4) + 4 = - 18 + 4$

Simplify the arithmetic:

$- 10 x = - 18 + 4$

Simplify the arithmetic:

$- 10 x = - 14$

Divide both sides by :

$\frac{(- 10 x)}{- 10} = \frac{- 14}{- 10}$

Cancel out the negatives:

$\frac{10 x}{10} = \frac{- 14}{- 10}$

Simplify the fraction:

$x = \frac{- 14}{- 10}$

Cancel out the negatives:

$x = \frac{14}{10}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

17 additional steps

$2 \cdot (x - 2) = 6 \cdot (- (2 x - 3))$

Expand the parentheses:

$2 x + 2 \cdot - 2 = 6 \cdot (- (2 x - 3))$

Simplify the arithmetic:

$2 x - 4 = 6 \cdot (- (2 x - 3))$

Expand the parentheses:

$2 x - 4 = 6 \cdot (- 2 x + 3)$

Expand the parentheses:

$2 x - 4 = 6 \cdot - 2 x + 6 \cdot 3$

Multiply the coefficients:

$2 x - 4 = - 12 x + 6 \cdot 3$

Simplify the arithmetic:

$2 x - 4 = - 12 x + 18$

Add to both sides:

$(2 x - 4) + 12 x = (- 12 x + 18) + 12 x$

Group like terms:

$(2 x + 12 x) - 4 = (- 12 x + 18) + 12 x$

Simplify the arithmetic:

$14 x - 4 = (- 12 x + 18) + 12 x$

Group like terms:

$14 x - 4 = (- 12 x + 12 x) + 18$

Simplify the arithmetic:

$14 x - 4 = 18$

Add to both sides:

$(14 x - 4) + 4 = 18 + 4$

Simplify the arithmetic:

$14 x = 18 + 4$

Simplify the arithmetic:

$14 x = 22$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{22}{14}$

Simplify the fraction:

$x = \frac{22}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(11 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{11}{7}$

3. List the solutions

$x = \frac{7}{5}, \frac{11}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | x - 2 |$
$y = 6 | 2 x - 3 |$
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

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