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Solution - Absolute value equations

Exact form:

x = - \frac{12}{5}, - \frac{5}{3}

x=-\frac{12}{5} , -\frac{5}{3}

Mixed number form:

x = - 2 \frac{2}{5}, - 1 \frac{2}{3}

x=-2\frac{2}{5} , -1\frac{2}{3}

Decimal form:

x = - 2.4, - 1.667

x=-2.4 , -1.667

See steps

Step-by-step explanation

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

$11 | x + 2 | - | x - 2 | = 0$

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

$11 | x + 2 | - | x - 2 | + | x - 2 | = | x - 2 |$

Simplify the arithmetic

$11 | x + 2 | = | x - 2 |$

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
$11 | x + 2 | = | x - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$11 \| x + 2 \| = \| x - 2 \|$
$x = + y$	$11 (x + 2) = (x - 2)$
$x = - y$	$11 (x + 2) = (- (x - 2))$
$+ x = y$	$11 (x + 2) = (x - 2)$
$- x = y$	$11 (- (x + 2)) = (x - 2)$

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:

$\| x \| = \| y \|$	$11 \| x + 2 \| = \| x - 2 \|$
$x = + y$ , $+ x = y$	$11 (x + 2) = (x - 2)$
$x = - y$ , $- x = y$	$11 (x + 2) = (- (x - 2))$

3. Solve the two equations for x

13 additional steps

$11 \cdot (x + 2) = (x - 2)$

Expand the parentheses:

$11 x + 11 \cdot 2 = (x - 2)$

Simplify the arithmetic:

$11 x + 22 = (x - 2)$

Subtract from both sides:

$(11 x + 22) - x = (x - 2) - x$

Group like terms:

$(11 x - x) + 22 = (x - 2) - x$

Simplify the arithmetic:

$10 x + 22 = (x - 2) - x$

Group like terms:

$10 x + 22 = (x - x) - 2$

Simplify the arithmetic:

$10 x + 22 = - 2$

Subtract from both sides:

$(10 x + 22) - 22 = - 2 - 22$

Simplify the arithmetic:

$10 x = - 2 - 22$

Simplify the arithmetic:

$10 x = - 24$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{- 24}{10}$

Simplify the fraction:

$x = \frac{- 24}{10}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 12 \cdot 2)}{(5 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 12}{5}$

14 additional steps

$11 \cdot (x + 2) = (- (x - 2))$

Expand the parentheses:

$11 x + 11 \cdot 2 = (- (x - 2))$

Simplify the arithmetic:

$11 x + 22 = (- (x - 2))$

Expand the parentheses:

$11 x + 22 = - x + 2$

Add to both sides:

$(11 x + 22) + x = (- x + 2) + x$

Group like terms:

$(11 x + x) + 22 = (- x + 2) + x$

Simplify the arithmetic:

$12 x + 22 = (- x + 2) + x$

Group like terms:

$12 x + 22 = (- x + x) + 2$

Simplify the arithmetic:

$12 x + 22 = 2$

Subtract from both sides:

$(12 x + 22) - 22 = 2 - 22$

Simplify the arithmetic:

$12 x = 2 - 22$

Simplify the arithmetic:

$12 x = - 20$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{- 20}{12}$

Simplify the fraction:

$x = \frac{- 20}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 4)}{(3 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 5}{3}$

4. List the solutions

$x = - \frac{12}{5}, - \frac{5}{3}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 11 | x + 2 |$
$y = | x - 2 |$
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 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