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Solution - Absolute value equations

Exact form:

x = 0, \frac{32}{5}

x=0 , \frac{32}{5}

Mixed number form:

x = 0, 6 \frac{2}{5}

x=0 , 6\frac{2}{5}

Decimal form:

x = 0, 6.4

x=0 , 6.4

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
$4 | x - 4 | = | x - 16 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| x - 4 \| = \| x - 16 \|$
$x = + y$	$4 (x - 4) = (x - 16)$
$x = - y$	$4 (x - 4) = - (x - 16)$
$+ x = y$	$4 (x - 4) = (x - 16)$
$- x = y$	$4 (- (x - 4)) = (x - 16)$

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:

$\| x \| = \| y \|$	$4 \| x - 4 \| = \| x - 16 \|$
$x = + y$ , $+ x = y$	$4 (x - 4) = (x - 16)$
$x = - y$ , $- x = y$	$4 (x - 4) = - (x - 16)$

2. Solve the two equations for x

10 additional steps

$4 \cdot (x - 4) = (x - 16)$

Expand the parentheses:

$4 x + 4 \cdot - 4 = (x - 16)$

Simplify the arithmetic:

$4 x - 16 = (x - 16)$

Subtract from both sides:

$(4 x - 16) - x = (x - 16) - x$

Group like terms:

$(4 x - x) - 16 = (x - 16) - x$

Simplify the arithmetic:

$3 x - 16 = (x - 16) - x$

Group like terms:

$3 x - 16 = (x - x) - 16$

Simplify the arithmetic:

$3 x - 16 = - 16$

Add to both sides:

$(3 x - 16) + 16 = - 16 + 16$

Simplify the arithmetic:

$3 x = - 16 + 16$

Simplify the arithmetic:

$3 x = 0$

Divide both sides by the coefficient:

$x = 0$

12 additional steps

$4 \cdot (x - 4) = - (x - 16)$

Expand the parentheses:

$4 x + 4 \cdot - 4 = - (x - 16)$

Simplify the arithmetic:

$4 x - 16 = - (x - 16)$

Expand the parentheses:

$4 x - 16 = - x + 16$

Add to both sides:

$(4 x - 16) + x = (- x + 16) + x$

Group like terms:

$(4 x + x) - 16 = (- x + 16) + x$

Simplify the arithmetic:

$5 x - 16 = (- x + 16) + x$

Group like terms:

$5 x - 16 = (- x + x) + 16$

Simplify the arithmetic:

$5 x - 16 = 16$

Add to both sides:

$(5 x - 16) + 16 = 16 + 16$

Simplify the arithmetic:

$5 x = 16 + 16$

Simplify the arithmetic:

$5 x = 32$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{32}{5}$

Simplify the fraction:

$x = \frac{32}{5}$

3. List the solutions

$x = 0, \frac{32}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 4 | x - 4 |$
$y = | x - 16 |$
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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