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Solution - Absolute value equations

Exact form:

x = - \frac{32}{3}, 0

x=-\frac{32}{3} , 0

Mixed number form:

x = - 10 \frac{2}{3}, 0

x=-10\frac{2}{3} , 0

Decimal form:

x = - 10.667, 0

x=-10.667 , 0

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

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

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:

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

2. Solve the two equations for x

13 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 3 x - 16 = (4 x + 16) - 4 x$

Group like terms:

$- 3 x - 16 = (4 x - 4 x) + 16$

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:

$- 3 x = 32$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{32}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{32}{- 3}$

Simplify the fraction:

$x = \frac{32}{- 3}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 32}{3}$

13 additional steps

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

Expand the parentheses:

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

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

Group like terms:

$(x - 16) = (4 \cdot - 1) x + 4 \cdot - 4$

Multiply the coefficients:

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

Simplify the arithmetic:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$5 x - 16 = (- 4 x - 16) + 4 x$

Group like terms:

$5 x - 16 = (- 4 x + 4 x) - 16$

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:

$5 x = 0$

Divide both sides by the coefficient:

$x = 0$

3. List the solutions

$x = - \frac{32}{3}, 0$
(2 solution(s))

4. Graph

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