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Solution - Absolute value equations

Exact form:

x = \frac{2}{3}, - \frac{4}{17}

x=\frac{2}{3} , -\frac{4}{17}

Decimal form:

x = 0.667, - 0.235

x=0.667 , -0.235

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
$| 4 x + 5 | = | 13 x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x + 5 \| = \| 13 x - 1 \|$
$x = + y$	$(4 x + 5) = (13 x - 1)$
$x = - y$	$(4 x + 5) = - (13 x - 1)$
$+ x = y$	$(4 x + 5) = (13 x - 1)$
$- x = y$	$- (4 x + 5) = (13 x - 1)$

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:

$\| x \| = \| y \|$	$\| 4 x + 5 \| = \| 13 x - 1 \|$
$x = + y$ , $+ x = y$	$(4 x + 5) = (13 x - 1)$
$x = - y$ , $- x = y$	$(4 x + 5) = - (13 x - 1)$

2. Solve the two equations for x

13 additional steps

$(4 x + 5) = (13 x - 1)$

Subtract from both sides:

$(4 x + 5) - 13 x = (13 x - 1) - 13 x$

Group like terms:

$(4 x - 13 x) + 5 = (13 x - 1) - 13 x$

Simplify the arithmetic:

$- 9 x + 5 = (13 x - 1) - 13 x$

Group like terms:

$- 9 x + 5 = (13 x - 13 x) - 1$

Simplify the arithmetic:

$- 9 x + 5 = - 1$

Subtract from both sides:

$(- 9 x + 5) - 5 = - 1 - 5$

Simplify the arithmetic:

$- 9 x = - 1 - 5$

Simplify the arithmetic:

$- 9 x = - 6$

Divide both sides by :

$\frac{(- 9 x)}{- 9} = \frac{- 6}{- 9}$

Cancel out the negatives:

$\frac{9 x}{9} = \frac{- 6}{- 9}$

Simplify the fraction:

$x = \frac{- 6}{- 9}$

Cancel out the negatives:

$x = \frac{6}{9}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 3)}{(3 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{2}{3}$

10 additional steps

$(4 x + 5) = - (13 x - 1)$

Expand the parentheses:

$(4 x + 5) = - 13 x + 1$

Add to both sides:

$(4 x + 5) + 13 x = (- 13 x + 1) + 13 x$

Group like terms:

$(4 x + 13 x) + 5 = (- 13 x + 1) + 13 x$

Simplify the arithmetic:

$17 x + 5 = (- 13 x + 1) + 13 x$

Group like terms:

$17 x + 5 = (- 13 x + 13 x) + 1$

Simplify the arithmetic:

$17 x + 5 = 1$

Subtract from both sides:

$(17 x + 5) - 5 = 1 - 5$

Simplify the arithmetic:

$17 x = 1 - 5$

Simplify the arithmetic:

$17 x = - 4$

Divide both sides by :

$\frac{(17 x)}{17} = \frac{- 4}{17}$

Simplify the fraction:

$x = \frac{- 4}{17}$

3. List the solutions

$x = \frac{2}{3}, - \frac{4}{17}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x + 5 |$
$y = | 13 x - 1 |$
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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