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Solution - Absolute value equations

Exact form:

a = 8, - 8

a=8 , -8

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
$| a - 8 | = | - a + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| a - 8 \| = \| - a + 8 \|$
$x = + y$	$(a - 8) = (- a + 8)$
$x = - y$	$(a - 8) = - (- a + 8)$
$+ x = y$	$(a - 8) = (- a + 8)$
$- x = y$	$- (a - 8) = (- a + 8)$

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:

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

2. Solve the two equations for a

11 additional steps

$(a - 8) = (- a + 8)$

Add to both sides:

$(a - 8) + a = (- a + 8) + a$

Group like terms:

$(a + a) - 8 = (- a + 8) + a$

Simplify the arithmetic:

$2 a - 8 = (- a + 8) + a$

Group like terms:

$2 a - 8 = (- a + a) + 8$

Simplify the arithmetic:

$2 a - 8 = 8$

Add to both sides:

$(2 a - 8) + 8 = 8 + 8$

Simplify the arithmetic:

$2 a = 8 + 8$

Simplify the arithmetic:

$2 a = 16$

Divide both sides by :

$\frac{(2 a)}{2} = \frac{16}{2}$

Simplify the fraction:

$a = \frac{16}{2}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(8 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = 8$

5 additional steps

$(a - 8) = - (- a + 8)$

Expand the parentheses:

$(a - 8) = a - 8$

Subtract from both sides:

$(a - 8) - a = (a - 8) - a$

Group like terms:

$(a - a) - 8 = (a - 8) - a$

Simplify the arithmetic:

$- 8 = (a - 8) - a$

Group like terms:

$- 8 = (a - a) - 8$

Simplify the arithmetic:

$- 8 = - 8$

3. List the solutions

$a = 8, - 8$
(2 solution(s))

4. Graph

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

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 a

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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