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Solution - Absolute value equations

Exact form:

x = - \frac{9}{4}, 9

x=-\frac{9}{4} , 9

Mixed number form:

x = - 2 \frac{1}{4}, 9

x=-2\frac{1}{4} , 9

Decimal form:

x = - 2.25, 9

x=-2.25 , 9

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

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

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:

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

2. Solve the two equations for x

11 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$8 x + 9 = (- 4 x - 9) + 4 x$

Group like terms:

$8 x + 9 = (- 4 x + 4 x) - 9$

Simplify the arithmetic:

$8 x + 9 = - 9$

Subtract from both sides:

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

Simplify the arithmetic:

$8 x = - 9 - 9$

Simplify the arithmetic:

$8 x = - 18$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{- 18}{8}$

Simplify the fraction:

$x = \frac{- 18}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 9 \cdot 2)}{(4 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 9}{4}$

5 additional steps

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

Expand the parentheses:

$(4 x + 9) = 4 x + 9$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$9 = 9$

3. List the solutions

$x = - \frac{9}{4}, 9$
(2 solution(s))

4. Graph

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