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Solution - Absolute value equations

Exact form:

x = \frac{1}{19}, - \frac{1}{20}

x=\frac{1}{19} , -\frac{1}{20}

Decimal form:

x = 0.053, - 0.05

x=0.053 , -0.05

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
$| 2 x + 4 | = | 78 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x + 4 \| = \| 78 x \|$
$x = + y$	$(2 x + 4) = (78 x)$
$x = - y$	$(2 x + 4) = - (78 x)$
$+ x = y$	$(2 x + 4) = (78 x)$
$- x = y$	$- (2 x + 4) = (78 x)$

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:

$\| x \| = \| y \|$	$\| 2 x + 4 \| = \| 78 x \|$
$x = + y$ , $+ x = y$	$(2 x + 4) = (78 x)$
$x = - y$ , $- x = y$	$(2 x + 4) = - (78 x)$

2. Solve the two equations for x

12 additional steps

$(2 x + 4) = 78 x$

Subtract from both sides:

$(2 x + 4) - 78 x = (78 x) - 78 x$

Group like terms:

$(2 x - 78 x) + 4 = (78 x) - 78 x$

Simplify the arithmetic:

$- 76 x + 4 = (78 x) - 78 x$

Simplify the arithmetic:

$- 76 x + 4 = 0$

Subtract from both sides:

$(- 76 x + 4) - 4 = 0 - 4$

Simplify the arithmetic:

$- 76 x = 0 - 4$

Simplify the arithmetic:

$- 76 x = - 4$

Divide both sides by :

$\frac{(- 76 x)}{- 76} = \frac{- 4}{- 76}$

Cancel out the negatives:

$\frac{76 x}{76} = \frac{- 4}{- 76}$

Simplify the fraction:

$x = \frac{- 4}{- 76}$

Cancel out the negatives:

$x = \frac{4}{76}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 4)}{(19 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{19}$

9 additional steps

$(2 x + 4) = - 78 x$

Subtract from both sides:

$(2 x + 4) - 4 = (- 78 x) - 4$

Simplify the arithmetic:

$2 x = (- 78 x) - 4$

Add to both sides:

$(2 x) + 78 x = ((- 78 x) - 4) + 78 x$

Simplify the arithmetic:

$80 x = ((- 78 x) - 4) + 78 x$

Group like terms:

$80 x = (- 78 x + 78 x) - 4$

Simplify the arithmetic:

$80 x = - 4$

Divide both sides by :

$\frac{(80 x)}{80} = \frac{- 4}{80}$

Simplify the fraction:

$x = \frac{- 4}{80}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 4)}{(20 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{20}$

3. List the solutions

$x = \frac{1}{19}, - \frac{1}{20}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x + 4 |$
$y = | 78 x |$
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 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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