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Solution - Absolute value equations

Exact form:

x = - \frac{1}{3}, 15

x=-\frac{1}{3} , 15

Decimal form:

x = - 0.333, 15

x=-0.333 , 15

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
$| 4 x + 9 | = | - 5 x + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x + 9 \| = \| - 5 x + 6 \|$
$x = + y$	$(4 x + 9) = (- 5 x + 6)$
$x = - y$	$(4 x + 9) = - (- 5 x + 6)$
$+ x = y$	$(4 x + 9) = (- 5 x + 6)$
$- x = y$	$- (4 x + 9) = (- 5 x + 6)$

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:

$\| x \| = \| y \|$	$\| 4 x + 9 \| = \| - 5 x + 6 \|$
$x = + y$ , $+ x = y$	$(4 x + 9) = (- 5 x + 6)$
$x = - y$ , $- x = y$	$(4 x + 9) = - (- 5 x + 6)$

2. Solve the two equations for x

11 additional steps

$(4 x + 9) = (- 5 x + 6)$

Add to both sides:

$(4 x + 9) + 5 x = (- 5 x + 6) + 5 x$

Group like terms:

$(4 x + 5 x) + 9 = (- 5 x + 6) + 5 x$

Simplify the arithmetic:

$9 x + 9 = (- 5 x + 6) + 5 x$

Group like terms:

$9 x + 9 = (- 5 x + 5 x) + 6$

Simplify the arithmetic:

$9 x + 9 = 6$

Subtract from both sides:

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

Simplify the arithmetic:

$9 x = 6 - 9$

Simplify the arithmetic:

$9 x = - 3$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{- 3}{9}$

Simplify the fraction:

$x = \frac{- 3}{9}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 3)}{(3 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{3}$

11 additional steps

$(4 x + 9) = - (- 5 x + 6)$

Expand the parentheses:

$(4 x + 9) = 5 x - 6$

Subtract from both sides:

$(4 x + 9) - 5 x = (5 x - 6) - 5 x$

Group like terms:

$(4 x - 5 x) + 9 = (5 x - 6) - 5 x$

Simplify the arithmetic:

$- x + 9 = (5 x - 6) - 5 x$

Group like terms:

$- x + 9 = (5 x - 5 x) - 6$

Simplify the arithmetic:

$- x + 9 = - 6$

Subtract from both sides:

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

Simplify the arithmetic:

$- x = - 6 - 9$

Simplify the arithmetic:

$- x = - 15$

Multiply both sides by :

$- x \cdot - 1 = - 15 \cdot - 1$

Remove the one(s):

$x = - 15 \cdot - 1$

Simplify the arithmetic:

$x = 15$

3. List the solutions

$x = - \frac{1}{3}, 15$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x + 9 |$
$y = | - 5 x + 6 |$
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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