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Solution - Absolute value equations

Exact form:

a = - \frac{7}{3}, - 1

a=-\frac{7}{3} , -1

Mixed number form:

a = - 2 \frac{1}{3}, - 1

a=-2\frac{1}{3} , -1

Decimal form:

a = - 2.333, - 1

a=-2.333 , -1

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
$| 4 a + 8 | = | - 2 a - 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 a + 8 \| = \| - 2 a - 6 \|$
$x = + y$	$(4 a + 8) = (- 2 a - 6)$
$x = - y$	$(4 a + 8) = - (- 2 a - 6)$
$+ x = y$	$(4 a + 8) = (- 2 a - 6)$
$- x = y$	$- (4 a + 8) = (- 2 a - 6)$

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:

$\| x \| = \| y \|$	$\| 4 a + 8 \| = \| - 2 a - 6 \|$
$x = + y$ , $+ x = y$	$(4 a + 8) = (- 2 a - 6)$
$x = - y$ , $- x = y$	$(4 a + 8) = - (- 2 a - 6)$

2. Solve the two equations for a

11 additional steps

$(4 a + 8) = (- 2 a - 6)$

Add to both sides:

$(4 a + 8) + 2 a = (- 2 a - 6) + 2 a$

Group like terms:

$(4 a + 2 a) + 8 = (- 2 a - 6) + 2 a$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$6 a + 8 = - 6$

Subtract from both sides:

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

Simplify the arithmetic:

$6 a = - 6 - 8$

Simplify the arithmetic:

$6 a = - 14$

Divide both sides by :

$\frac{(6 a)}{6} = \frac{- 14}{6}$

Simplify the fraction:

$a = \frac{- 14}{6}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(- 7 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = \frac{- 7}{3}$

11 additional steps

$(4 a + 8) = - (- 2 a - 6)$

Expand the parentheses:

$(4 a + 8) = 2 a + 6$

Subtract from both sides:

$(4 a + 8) - 2 a = (2 a + 6) - 2 a$

Group like terms:

$(4 a - 2 a) + 8 = (2 a + 6) - 2 a$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 a + 8 = 6$

Subtract from both sides:

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

Simplify the arithmetic:

$2 a = 6 - 8$

Simplify the arithmetic:

$2 a = - 2$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$a = - 1$

3. List the solutions

$a = - \frac{7}{3}, - 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 a + 8 |$
$y = | - 2 a - 6 |$
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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