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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 0.067, 3

x=-0.067 , 3

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

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

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:

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

2. Solve the two equations for x

11 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 15 x - 5 = (9 x - 4) - 9 x$

Group like terms:

$- 15 x - 5 = (9 x - 9 x) - 4$

Simplify the arithmetic:

$- 15 x - 5 = - 4$

Add to both sides:

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

Simplify the arithmetic:

$- 15 x = - 4 + 5$

Simplify the arithmetic:

$- 15 x = 1$

Divide both sides by :

$\frac{(- 15 x)}{- 15} = \frac{1}{- 15}$

Cancel out the negatives:

$\frac{15 x}{15} = \frac{1}{- 15}$

Simplify the fraction:

$x = \frac{1}{- 15}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 1}{15}$

12 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$3 x - 5 = (- 9 x + 4) + 9 x$

Group like terms:

$3 x - 5 = (- 9 x + 9 x) + 4$

Simplify the arithmetic:

$3 x - 5 = 4$

Add to both sides:

$(3 x - 5) + 5 = 4 + 5$

Simplify the arithmetic:

$3 x = 4 + 5$

Simplify the arithmetic:

$3 x = 9$

Divide both sides by :

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

Simplify the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = 3$

3. List the solutions

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

4. Graph

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