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Solution - Absolute value equations

Exact form:

a = \frac{7}{3}

a=\frac{7}{3}

Mixed number form:

a = 2 \frac{1}{3}

a=2\frac{1}{3}

Decimal form:

a = 2.333

a=2.333

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
$| - 3 a + 6 | = | - 3 a + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 a + 6 \| = \| - 3 a + 8 \|$
$x = + y$	$(- 3 a + 6) = (- 3 a + 8)$
$x = - y$	$(- 3 a + 6) = - (- 3 a + 8)$
$+ x = y$	$(- 3 a + 6) = (- 3 a + 8)$
$- x = y$	$- (- 3 a + 6) = (- 3 a + 8)$

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:

$\| x \| = \| y \|$	$\| - 3 a + 6 \| = \| - 3 a + 8 \|$
$x = + y$ , $+ x = y$	$(- 3 a + 6) = (- 3 a + 8)$
$x = - y$ , $- x = y$	$(- 3 a + 6) = - (- 3 a + 8)$

2. Solve the two equations for a

5 additional steps

$(- 3 a + 6) = (- 3 a + 8)$

Add to both sides:

$(- 3 a + 6) + 3 a = (- 3 a + 8) + 3 a$

Group like terms:

$(- 3 a + 3 a) + 6 = (- 3 a + 8) + 3 a$

Simplify the arithmetic:

$6 = (- 3 a + 8) + 3 a$

Group like terms:

$6 = (- 3 a + 3 a) + 8$

Simplify the arithmetic:

$6 = 8$

The statement is false:

$6 = 8$

The equation is false so it has no solution.

14 additional steps

$(- 3 a + 6) = - (- 3 a + 8)$

Expand the parentheses:

$(- 3 a + 6) = 3 a - 8$

Subtract from both sides:

$(- 3 a + 6) - 3 a = (3 a - 8) - 3 a$

Group like terms:

$(- 3 a - 3 a) + 6 = (3 a - 8) - 3 a$

Simplify the arithmetic:

$- 6 a + 6 = (3 a - 8) - 3 a$

Group like terms:

$- 6 a + 6 = (3 a - 3 a) - 8$

Simplify the arithmetic:

$- 6 a + 6 = - 8$

Subtract from both sides:

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

Simplify the arithmetic:

$- 6 a = - 8 - 6$

Simplify the arithmetic:

$- 6 a = - 14$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

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:

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

3. Graph

Each line represents the function of one side of the equation:
$y = | - 3 a + 6 |$
$y = | - 3 a + 8 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved