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Solution - Absolute value equations

Exact form:

x = - \frac{1}{2}, - 4

x=-\frac{1}{2} , -4

Decimal form:

x = - 0.5, - 4

x=-0.5 , -4

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
$| - 3 x - 5 | = | x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 x - 5 \| = \| x - 3 \|$
$x = + y$	$(- 3 x - 5) = (x - 3)$
$x = - y$	$(- 3 x - 5) = - (x - 3)$
$+ x = y$	$(- 3 x - 5) = (x - 3)$
$- x = y$	$- (- 3 x - 5) = (x - 3)$

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:

$\| x \| = \| y \|$	$\| - 3 x - 5 \| = \| x - 3 \|$
$x = + y$ , $+ x = y$	$(- 3 x - 5) = (x - 3)$
$x = - y$ , $- x = y$	$(- 3 x - 5) = - (x - 3)$

2. Solve the two equations for x

13 additional steps

$(- 3 x - 5) = (x - 3)$

Subtract from both sides:

$(- 3 x - 5) - x = (x - 3) - x$

Group like terms:

$(- 3 x - x) - 5 = (x - 3) - x$

Simplify the arithmetic:

$- 4 x - 5 = (x - 3) - x$

Group like terms:

$- 4 x - 5 = (x - x) - 3$

Simplify the arithmetic:

$- 4 x - 5 = - 3$

Add to both sides:

$(- 4 x - 5) + 5 = - 3 + 5$

Simplify the arithmetic:

$- 4 x = - 3 + 5$

Simplify the arithmetic:

$- 4 x = 2$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

14 additional steps

$(- 3 x - 5) = - (x - 3)$

Expand the parentheses:

$(- 3 x - 5) = - x + 3$

Add to both sides:

$(- 3 x - 5) + x = (- x + 3) + x$

Group like terms:

$(- 3 x + x) - 5 = (- x + 3) + x$

Simplify the arithmetic:

$- 2 x - 5 = (- x + 3) + x$

Group like terms:

$- 2 x - 5 = (- x + x) + 3$

Simplify the arithmetic:

$- 2 x - 5 = 3$

Add to both sides:

$(- 2 x - 5) + 5 = 3 + 5$

Simplify the arithmetic:

$- 2 x = 3 + 5$

Simplify the arithmetic:

$- 2 x = 8$

Divide both sides by :

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

Cancel out the negatives:

$\frac{2 x}{2} = \frac{8}{- 2}$

Simplify the fraction:

$x = \frac{8}{- 2}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 8}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = - 4$

3. List the solutions

$x = - \frac{1}{2}, - 4$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 x - 5 |$
$y = | x - 3 |$
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 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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