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Solution - Absolute value equations

Exact form:

x = - \frac{2}{3}, 1

x=-\frac{2}{3} , 1

Decimal form:

x = - 0.667, 1

x=-0.667 , 1

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 2 x + 8 | + | 10 x | = 0$

Add $- | 10 x |$ to both sides of the equation:

$| 2 x + 8 | + | 10 x | - | 10 x | = - | 10 x |$

Simplify the arithmetic

$| 2 x + 8 | = - | 10 x |$

2. 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
$| 2 x + 8 | = - | 10 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x + 8 \| = - \| 10 x \|$
$x = + y$	$(2 x + 8) = - (10 x)$
$x = - y$	$(2 x + 8) = - - (10 x)$
$+ x = y$	$(2 x + 8) = - (10 x)$
$- x = y$	$- (2 x + 8) = - (10 x)$

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:

$\| x \| = \| y \|$	$\| 2 x + 8 \| = - \| 10 x \|$
$x = + y$ , $+ x = y$	$(2 x + 8) = - (10 x)$
$x = - y$ , $- x = y$	$(2 x + 8) = - - (10 x)$

3. Solve the two equations for x

9 additional steps

$(2 x + 8) = - 10 x$

Subtract from both sides:

$(2 x + 8) - 8 = (- 10 x) - 8$

Simplify the arithmetic:

$2 x = (- 10 x) - 8$

Add to both sides:

$(2 x) + 10 x = ((- 10 x) - 8) + 10 x$

Simplify the arithmetic:

$12 x = ((- 10 x) - 8) + 10 x$

Group like terms:

$12 x = (- 10 x + 10 x) - 8$

Simplify the arithmetic:

$12 x = - 8$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{- 8}{12}$

Simplify the fraction:

$x = \frac{- 8}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 4)}{(3 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 2}{3}$

13 additional steps

$(2 x + 8) = - - 10 x$

Group like terms:

$(2 x + 8) = (- 1 \cdot - 10) x$

Multiply the coefficients:

$(2 x + 8) = 10 x$

Subtract from both sides:

$(2 x + 8) - 10 x = (10 x) - 10 x$

Group like terms:

$(2 x - 10 x) + 8 = (10 x) - 10 x$

Simplify the arithmetic:

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

Simplify the arithmetic:

$- 8 x + 8 = 0$

Subtract from both sides:

$(- 8 x + 8) - 8 = 0 - 8$

Simplify the arithmetic:

$- 8 x = 0 - 8$

Simplify the arithmetic:

$- 8 x = - 8$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

Simplify the fraction:

$x = 1$

4. List the solutions

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

5. Graph

Each line represents the function of one side of the equation:
$y = | 2 x + 8 |$
$y = - | 10 x |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved