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Solution - Absolute value equations

Exact form:

a = 10, \frac{2}{3}

a=10 , \frac{2}{3}

Decimal form:

a = 10, 0.667

a=10 , 0.667

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
$4 | a - 3 | = | 2 a + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| a - 3 \| = \| 2 a + 8 \|$
$x = + y$	$4 (a - 3) = (2 a + 8)$
$x = - y$	$4 (a - 3) = - (2 a + 8)$
$+ x = y$	$4 (a - 3) = (2 a + 8)$
$- x = y$	$4 (- (a - 3)) = (2 a + 8)$

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:

$\| x \| = \| y \|$	$4 \| a - 3 \| = \| 2 a + 8 \|$
$x = + y$ , $+ x = y$	$4 (a - 3) = (2 a + 8)$
$x = - y$ , $- x = y$	$4 (a - 3) = - (2 a + 8)$

2. Solve the two equations for a

13 additional steps

$4 \cdot (a - 3) = (2 a + 8)$

Expand the parentheses:

$4 a + 4 \cdot - 3 = (2 a + 8)$

Simplify the arithmetic:

$4 a - 12 = (2 a + 8)$

Subtract from both sides:

$(4 a - 12) - 2 a = (2 a + 8) - 2 a$

Group like terms:

$(4 a - 2 a) - 12 = (2 a + 8) - 2 a$

Simplify the arithmetic:

$2 a - 12 = (2 a + 8) - 2 a$

Group like terms:

$2 a - 12 = (2 a - 2 a) + 8$

Simplify the arithmetic:

$2 a - 12 = 8$

Add to both sides:

$(2 a - 12) + 12 = 8 + 12$

Simplify the arithmetic:

$2 a = 8 + 12$

Simplify the arithmetic:

$2 a = 20$

Divide both sides by :

$\frac{(2 a)}{2} = \frac{20}{2}$

Simplify the fraction:

$a = \frac{20}{2}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(10 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = 10$

14 additional steps

$4 \cdot (a - 3) = - (2 a + 8)$

Expand the parentheses:

$4 a + 4 \cdot - 3 = - (2 a + 8)$

Simplify the arithmetic:

$4 a - 12 = - (2 a + 8)$

Expand the parentheses:

$4 a - 12 = - 2 a - 8$

Add to both sides:

$(4 a - 12) + 2 a = (- 2 a - 8) + 2 a$

Group like terms:

$(4 a + 2 a) - 12 = (- 2 a - 8) + 2 a$

Simplify the arithmetic:

$6 a - 12 = (- 2 a - 8) + 2 a$

Group like terms:

$6 a - 12 = (- 2 a + 2 a) - 8$

Simplify the arithmetic:

$6 a - 12 = - 8$

Add to both sides:

$(6 a - 12) + 12 = - 8 + 12$

Simplify the arithmetic:

$6 a = - 8 + 12$

Simplify the arithmetic:

$6 a = 4$

Divide both sides by :

$\frac{(6 a)}{6} = \frac{4}{6}$

Simplify the fraction:

$a = \frac{4}{6}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(2 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = \frac{2}{3}$

3. List the solutions

$a = 10, \frac{2}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 4 | a - 3 |$
$y = | 2 a + 8 |$
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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