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Solution - Absolute value equations

Exact form:

x = - 4, \frac{8}{7}

x=-4 , \frac{8}{7}

Mixed number form:

x = - 4, 1 \frac{1}{7}

x=-4 , 1\frac{1}{7}

Decimal form:

x = - 4, 1.143

x=-4 , 1.143

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
$| 4 x - 20 | = | 10 x + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 20 \| = \| 10 x + 4 \|$
$x = + y$	$(4 x - 20) = (10 x + 4)$
$x = - y$	$(4 x - 20) = - (10 x + 4)$
$+ x = y$	$(4 x - 20) = (10 x + 4)$
$- x = y$	$- (4 x - 20) = (10 x + 4)$

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:

$\| x \| = \| y \|$	$\| 4 x - 20 \| = \| 10 x + 4 \|$
$x = + y$ , $+ x = y$	$(4 x - 20) = (10 x + 4)$
$x = - y$ , $- x = y$	$(4 x - 20) = - (10 x + 4)$

2. Solve the two equations for x

13 additional steps

$(4 x - 20) = (10 x + 4)$

Subtract from both sides:

$(4 x - 20) - 10 x = (10 x + 4) - 10 x$

Group like terms:

$(4 x - 10 x) - 20 = (10 x + 4) - 10 x$

Simplify the arithmetic:

$- 6 x - 20 = (10 x + 4) - 10 x$

Group like terms:

$- 6 x - 20 = (10 x - 10 x) + 4$

Simplify the arithmetic:

$- 6 x - 20 = 4$

Add to both sides:

$(- 6 x - 20) + 20 = 4 + 20$

Simplify the arithmetic:

$- 6 x = 4 + 20$

Simplify the arithmetic:

$- 6 x = 24$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{24}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{24}{- 6}$

Simplify the fraction:

$x = \frac{24}{- 6}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 24}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 4 \cdot 6)}{(1 \cdot 6)}$

Factor out and cancel the greatest common factor:

$x = - 4$

12 additional steps

$(4 x - 20) = - (10 x + 4)$

Expand the parentheses:

$(4 x - 20) = - 10 x - 4$

Add to both sides:

$(4 x - 20) + 10 x = (- 10 x - 4) + 10 x$

Group like terms:

$(4 x + 10 x) - 20 = (- 10 x - 4) + 10 x$

Simplify the arithmetic:

$14 x - 20 = (- 10 x - 4) + 10 x$

Group like terms:

$14 x - 20 = (- 10 x + 10 x) - 4$

Simplify the arithmetic:

$14 x - 20 = - 4$

Add to both sides:

$(14 x - 20) + 20 = - 4 + 20$

Simplify the arithmetic:

$14 x = - 4 + 20$

Simplify the arithmetic:

$14 x = 16$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{16}{14}$

Simplify the fraction:

$x = \frac{16}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(8 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{8}{7}$

3. List the solutions

$x = - 4, \frac{8}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 20 |$
$y = | 10 x + 4 |$
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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