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Solution - Absolute value equations

Exact form:

a = 0, 0

a=0 , 0

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
$| 3 a | = | \frac{1}{3} a |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 a \| = \| \frac{1}{3} a \|$
$x = + y$	$(3 a) = (\frac{1}{3} a)$
$x = - y$	$(3 a) = - (\frac{1}{3} a)$
$+ x = y$	$(3 a) = (\frac{1}{3} a)$
$- x = y$	$- (3 a) = (\frac{1}{3} a)$

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:

$\| x \| = \| y \|$	$\| 3 a \| = \| \frac{1}{3} a \|$
$x = + y$ , $+ x = y$	$(3 a) = (\frac{1}{3} a)$
$x = - y$ , $- x = y$	$(3 a) = - (\frac{1}{3} a)$

2. Solve the two equations for a

9 additional steps

$3 a = \frac{1}{3} a$

Subtract from both sides:

$(3 a) - \frac{1}{3} \cdot a = (\frac{1}{3} a) - \frac{1}{3} a$

Group the coefficients:

$(3 + \frac{- 1}{3}) a = (\frac{1}{3} \cdot a) - \frac{1}{3} a$

Convert the integer into a fraction:

$(\frac{9}{3} + \frac{- 1}{3}) a = (\frac{1}{3} \cdot a) - \frac{1}{3} a$

Combine the fractions:

$\frac{(9 - 1)}{3} \cdot a = (\frac{1}{3} \cdot a) - \frac{1}{3} a$

Combine the numerators:

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

Combine the fractions:

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

Combine the numerators:

$\frac{8}{3} \cdot a = \frac{0}{3} a$

Reduce the zero numerator:

$\frac{8}{3} a = 0 a$

Simplify the arithmetic:

$\frac{8}{3} a = 0$

Divide both sides by the coefficient:

$a = 0$

$3 a = \frac{- 1}{3} a$

Divide both sides by :

$\frac{(3 a)}{3} = \frac{(\frac{- 1}{3} a)}{3}$

Simplify the fraction:

$a = \frac{(\frac{- 1}{3} a)}{3}$

3. List the solutions

$a = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 a |$
$y = | \frac{1}{3} a |$
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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