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Solution - Absolute value equations

Exact form:

x = 3

x=3

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
$| 4 x - 16 | = | - 4 x + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 16 \| = \| - 4 x + 8 \|$
$x = + y$	$(4 x - 16) = (- 4 x + 8)$
$x = - y$	$(4 x - 16) = - (- 4 x + 8)$
$+ x = y$	$(4 x - 16) = (- 4 x + 8)$
$- x = y$	$- (4 x - 16) = (- 4 x + 8)$

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:

$\| x \| = \| y \|$	$\| 4 x - 16 \| = \| - 4 x + 8 \|$
$x = + y$ , $+ x = y$	$(4 x - 16) = (- 4 x + 8)$
$x = - y$ , $- x = y$	$(4 x - 16) = - (- 4 x + 8)$

2. Solve the two equations for x

11 additional steps

$(4 x - 16) = (- 4 x + 8)$

Add to both sides:

$(4 x - 16) + 4 x = (- 4 x + 8) + 4 x$

Group like terms:

$(4 x + 4 x) - 16 = (- 4 x + 8) + 4 x$

Simplify the arithmetic:

$8 x - 16 = (- 4 x + 8) + 4 x$

Group like terms:

$8 x - 16 = (- 4 x + 4 x) + 8$

Simplify the arithmetic:

$8 x - 16 = 8$

Add to both sides:

$(8 x - 16) + 16 = 8 + 16$

Simplify the arithmetic:

$8 x = 8 + 16$

Simplify the arithmetic:

$8 x = 24$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{24}{8}$

Simplify the fraction:

$x = \frac{24}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 8)}{(1 \cdot 8)}$

Factor out and cancel the greatest common factor:

$x = 3$

6 additional steps

$(4 x - 16) = - (- 4 x + 8)$

Expand the parentheses:

$(4 x - 16) = 4 x - 8$

Subtract from both sides:

$(4 x - 16) - 4 x = (4 x - 8) - 4 x$

Group like terms:

$(4 x - 4 x) - 16 = (4 x - 8) - 4 x$

Simplify the arithmetic:

$- 16 = (4 x - 8) - 4 x$

Group like terms:

$- 16 = (4 x - 4 x) - 8$

Simplify the arithmetic:

$- 16 = - 8$

The statement is false:

$- 16 = - 8$

The equation is false so it has no solution.

3. List the solutions

$x = 3$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 16 |$
$y = | - 4 x + 8 |$
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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