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Solution - Absolute value equations

Exact form:

x = - \frac{17}{4}, - \frac{1}{16}

x=-\frac{17}{4} , -\frac{1}{16}

Mixed number form:

x = - 4 \frac{1}{4}, - \frac{1}{16}

x=-4\frac{1}{4} , -\frac{1}{16}

Decimal form:

x = - 4.25, - 0.062

x=-4.25 , -0.062

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
$| 6 x - 8 | = | 10 x + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 x - 8 \| = \| 10 x + 9 \|$
$x = + y$	$(6 x - 8) = (10 x + 9)$
$x = - y$	$(6 x - 8) = - (10 x + 9)$
$+ x = y$	$(6 x - 8) = (10 x + 9)$
$- x = y$	$- (6 x - 8) = (10 x + 9)$

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:

$\| x \| = \| y \|$	$\| 6 x - 8 \| = \| 10 x + 9 \|$
$x = + y$ , $+ x = y$	$(6 x - 8) = (10 x + 9)$
$x = - y$ , $- x = y$	$(6 x - 8) = - (10 x + 9)$

2. Solve the two equations for x

11 additional steps

$(6 x - 8) = (10 x + 9)$

Subtract from both sides:

$(6 x - 8) - 10 x = (10 x + 9) - 10 x$

Group like terms:

$(6 x - 10 x) - 8 = (10 x + 9) - 10 x$

Simplify the arithmetic:

$- 4 x - 8 = (10 x + 9) - 10 x$

Group like terms:

$- 4 x - 8 = (10 x - 10 x) + 9$

Simplify the arithmetic:

$- 4 x - 8 = 9$

Add to both sides:

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

Simplify the arithmetic:

$- 4 x = 9 + 8$

Simplify the arithmetic:

$- 4 x = 17$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{17}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{17}{- 4}$

Simplify the fraction:

$x = \frac{17}{- 4}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 17}{4}$

10 additional steps

$(6 x - 8) = - (10 x + 9)$

Expand the parentheses:

$(6 x - 8) = - 10 x - 9$

Add to both sides:

$(6 x - 8) + 10 x = (- 10 x - 9) + 10 x$

Group like terms:

$(6 x + 10 x) - 8 = (- 10 x - 9) + 10 x$

Simplify the arithmetic:

$16 x - 8 = (- 10 x - 9) + 10 x$

Group like terms:

$16 x - 8 = (- 10 x + 10 x) - 9$

Simplify the arithmetic:

$16 x - 8 = - 9$

Add to both sides:

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

Simplify the arithmetic:

$16 x = - 9 + 8$

Simplify the arithmetic:

$16 x = - 1$

Divide both sides by :

$\frac{(16 x)}{16} = \frac{- 1}{16}$

Simplify the fraction:

$x = \frac{- 1}{16}$

3. List the solutions

$x = - \frac{17}{4}, - \frac{1}{16}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 x - 8 |$
$y = | 10 x + 9 |$
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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