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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = 5, - 0.714

x=5 , -0.714

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
$| 9 x - 5 | = | 5 x + 15 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x - 5 \| = \| 5 x + 15 \|$
$x = + y$	$(9 x - 5) = (5 x + 15)$
$x = - y$	$(9 x - 5) = - (5 x + 15)$
$+ x = y$	$(9 x - 5) = (5 x + 15)$
$- x = y$	$- (9 x - 5) = (5 x + 15)$

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:

$\| x \| = \| y \|$	$\| 9 x - 5 \| = \| 5 x + 15 \|$
$x = + y$ , $+ x = y$	$(9 x - 5) = (5 x + 15)$
$x = - y$ , $- x = y$	$(9 x - 5) = - (5 x + 15)$

2. Solve the two equations for x

11 additional steps

$(9 x - 5) = (5 x + 15)$

Subtract from both sides:

$(9 x - 5) - 5 x = (5 x + 15) - 5 x$

Group like terms:

$(9 x - 5 x) - 5 = (5 x + 15) - 5 x$

Simplify the arithmetic:

$4 x - 5 = (5 x + 15) - 5 x$

Group like terms:

$4 x - 5 = (5 x - 5 x) + 15$

Simplify the arithmetic:

$4 x - 5 = 15$

Add to both sides:

$(4 x - 5) + 5 = 15 + 5$

Simplify the arithmetic:

$4 x = 15 + 5$

Simplify the arithmetic:

$4 x = 20$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{20}{4}$

Simplify the fraction:

$x = \frac{20}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = 5$

12 additional steps

$(9 x - 5) = - (5 x + 15)$

Expand the parentheses:

$(9 x - 5) = - 5 x - 15$

Add to both sides:

$(9 x - 5) + 5 x = (- 5 x - 15) + 5 x$

Group like terms:

$(9 x + 5 x) - 5 = (- 5 x - 15) + 5 x$

Simplify the arithmetic:

$14 x - 5 = (- 5 x - 15) + 5 x$

Group like terms:

$14 x - 5 = (- 5 x + 5 x) - 15$

Simplify the arithmetic:

$14 x - 5 = - 15$

Add to both sides:

$(14 x - 5) + 5 = - 15 + 5$

Simplify the arithmetic:

$14 x = - 15 + 5$

Simplify the arithmetic:

$14 x = - 10$

Divide both sides by :

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

Simplify the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x - 5 |$
$y = | 5 x + 15 |$
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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