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Solution - Absolute value equations

Exact form:

x = - \frac{5}{31}, \frac{5}{33}

x=-\frac{5}{31} , \frac{5}{33}

Decimal form:

x = - 0.161, 0.152

x=-0.161 , 0.152

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
$| x - 5 | = | 32 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 5 \| = \| 32 x \|$
$x = + y$	$(x - 5) = (32 x)$
$x = - y$	$(x - 5) = - (32 x)$
$+ x = y$	$(x - 5) = (32 x)$
$- x = y$	$- (x - 5) = (32 x)$

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:

$\| x \| = \| y \|$	$\| x - 5 \| = \| 32 x \|$
$x = + y$ , $+ x = y$	$(x - 5) = (32 x)$
$x = - y$ , $- x = y$	$(x - 5) = - (32 x)$

2. Solve the two equations for x

10 additional steps

$(x - 5) = 32 x$

Subtract from both sides:

$(x - 5) - 32 x = (32 x) - 32 x$

Group like terms:

$(x - 32 x) - 5 = (32 x) - 32 x$

Simplify the arithmetic:

$- 31 x - 5 = (32 x) - 32 x$

Simplify the arithmetic:

$- 31 x - 5 = 0$

Add to both sides:

$(- 31 x - 5) + 5 = 0 + 5$

Simplify the arithmetic:

$- 31 x = 0 + 5$

Simplify the arithmetic:

$- 31 x = 5$

Divide both sides by :

$\frac{(- 31 x)}{- 31} = \frac{5}{- 31}$

Cancel out the negatives:

$\frac{31 x}{31} = \frac{5}{- 31}$

Simplify the fraction:

$x = \frac{5}{- 31}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 5}{31}$

7 additional steps

$(x - 5) = - 32 x$

Add to both sides:

$(x - 5) + 5 = (- 32 x) + 5$

Simplify the arithmetic:

$x = (- 32 x) + 5$

Add to both sides:

$x + 32 x = ((- 32 x) + 5) + 32 x$

Simplify the arithmetic:

$33 x = ((- 32 x) + 5) + 32 x$

Group like terms:

$33 x = (- 32 x + 32 x) + 5$

Simplify the arithmetic:

$33 x = 5$

Divide both sides by :

$\frac{(33 x)}{33} = \frac{5}{33}$

Simplify the fraction:

$x = \frac{5}{33}$

3. List the solutions

$x = - \frac{5}{31}, \frac{5}{33}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 5 |$
$y = | 32 x |$
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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