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Solution - Absolute value equations

Exact form:

x = 6, - \frac{4}{7}

x=6 , -\frac{4}{7}

Decimal form:

x = 6, - 0.571

x=6 , -0.571

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
$| 9 x - 8 | = | 5 x + 16 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x - 8 \| = \| 5 x + 16 \|$
$x = + y$	$(9 x - 8) = (5 x + 16)$
$x = - y$	$(9 x - 8) = - (5 x + 16)$
$+ x = y$	$(9 x - 8) = (5 x + 16)$
$- x = y$	$- (9 x - 8) = (5 x + 16)$

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:

$\| x \| = \| y \|$	$\| 9 x - 8 \| = \| 5 x + 16 \|$
$x = + y$ , $+ x = y$	$(9 x - 8) = (5 x + 16)$
$x = - y$ , $- x = y$	$(9 x - 8) = - (5 x + 16)$

2. Solve the two equations for x

11 additional steps

$(9 x - 8) = (5 x + 16)$

Subtract from both sides:

$(9 x - 8) - 5 x = (5 x + 16) - 5 x$

Group like terms:

$(9 x - 5 x) - 8 = (5 x + 16) - 5 x$

Simplify the arithmetic:

$4 x - 8 = (5 x + 16) - 5 x$

Group like terms:

$4 x - 8 = (5 x - 5 x) + 16$

Simplify the arithmetic:

$4 x - 8 = 16$

Add to both sides:

$(4 x - 8) + 8 = 16 + 8$

Simplify the arithmetic:

$4 x = 16 + 8$

Simplify the arithmetic:

$4 x = 24$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{24}{4}$

Simplify the fraction:

$x = \frac{24}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(6 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = 6$

12 additional steps

$(9 x - 8) = - (5 x + 16)$

Expand the parentheses:

$(9 x - 8) = - 5 x - 16$

Add to both sides:

$(9 x - 8) + 5 x = (- 5 x - 16) + 5 x$

Group like terms:

$(9 x + 5 x) - 8 = (- 5 x - 16) + 5 x$

Simplify the arithmetic:

$14 x - 8 = (- 5 x - 16) + 5 x$

Group like terms:

$14 x - 8 = (- 5 x + 5 x) - 16$

Simplify the arithmetic:

$14 x - 8 = - 16$

Add to both sides:

$(14 x - 8) + 8 = - 16 + 8$

Simplify the arithmetic:

$14 x = - 16 + 8$

Simplify the arithmetic:

$14 x = - 8$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{- 8}{14}$

Simplify the fraction:

$x = \frac{- 8}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 4 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 4}{7}$

3. List the solutions

$x = 6, - \frac{4}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x - 8 |$
$y = | 5 x + 16 |$
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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