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Solution - Absolute value equations

Exact form:

x = 16, - \frac{7}{5}

x=16 , -\frac{7}{5}

Mixed number form:

x = 16, - 1 \frac{2}{5}

x=16 , -1\frac{2}{5}

Decimal form:

x = 16, - 1.4

x=16 , -1.4

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
$| 4 x + 23 | = | 6 x - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x + 23 \| = \| 6 x - 9 \|$
$x = + y$	$(4 x + 23) = (6 x - 9)$
$x = - y$	$(4 x + 23) = - (6 x - 9)$
$+ x = y$	$(4 x + 23) = (6 x - 9)$
$- x = y$	$- (4 x + 23) = (6 x - 9)$

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:

$\| x \| = \| y \|$	$\| 4 x + 23 \| = \| 6 x - 9 \|$
$x = + y$ , $+ x = y$	$(4 x + 23) = (6 x - 9)$
$x = - y$ , $- x = y$	$(4 x + 23) = - (6 x - 9)$

2. Solve the two equations for x

13 additional steps

$(4 x + 23) = (6 x - 9)$

Subtract from both sides:

$(4 x + 23) - 6 x = (6 x - 9) - 6 x$

Group like terms:

$(4 x - 6 x) + 23 = (6 x - 9) - 6 x$

Simplify the arithmetic:

$- 2 x + 23 = (6 x - 9) - 6 x$

Group like terms:

$- 2 x + 23 = (6 x - 6 x) - 9$

Simplify the arithmetic:

$- 2 x + 23 = - 9$

Subtract from both sides:

$(- 2 x + 23) - 23 = - 9 - 23$

Simplify the arithmetic:

$- 2 x = - 9 - 23$

Simplify the arithmetic:

$- 2 x = - 32$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{- 32}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{- 32}{- 2}$

Simplify the fraction:

$x = \frac{- 32}{- 2}$

Cancel out the negatives:

$x = \frac{32}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(16 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 16$

12 additional steps

$(4 x + 23) = - (6 x - 9)$

Expand the parentheses:

$(4 x + 23) = - 6 x + 9$

Add to both sides:

$(4 x + 23) + 6 x = (- 6 x + 9) + 6 x$

Group like terms:

$(4 x + 6 x) + 23 = (- 6 x + 9) + 6 x$

Simplify the arithmetic:

$10 x + 23 = (- 6 x + 9) + 6 x$

Group like terms:

$10 x + 23 = (- 6 x + 6 x) + 9$

Simplify the arithmetic:

$10 x + 23 = 9$

Subtract from both sides:

$(10 x + 23) - 23 = 9 - 23$

Simplify the arithmetic:

$10 x = 9 - 23$

Simplify the arithmetic:

$10 x = - 14$

Divide both sides by :

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

Simplify the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$x = 16, - \frac{7}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x + 23 |$
$y = | 6 x - 9 |$
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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