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Solution - Absolute value equations

Exact form:

x = 0, 0

x=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
$| \frac{1}{3} x | = | \frac{2}{5} x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{1}{3} x \| = \| \frac{2}{5} x \|$
$x = + y$	$(\frac{1}{3} x) = (\frac{2}{5} x)$
$x = - y$	$(\frac{1}{3} x) = - (\frac{2}{5} x)$
$+ x = y$	$(\frac{1}{3} x) = (\frac{2}{5} x)$
$- x = y$	$- (\frac{1}{3} x) = (\frac{2}{5} x)$

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:

$\| x \| = \| y \|$	$\| \frac{1}{3} x \| = \| \frac{2}{5} x \|$
$x = + y$ , $+ x = y$	$(\frac{1}{3} x) = (\frac{2}{5} x)$
$x = - y$ , $- x = y$	$(\frac{1}{3} x) = - (\frac{2}{5} x)$

2. Solve the two equations for x

11 additional steps

$\frac{1}{3} \cdot x = \frac{2}{5} x$

Subtract from both sides:

$(\frac{1}{3} x) - \frac{2}{5} \cdot x = (\frac{2}{5} x) - \frac{2}{5} x$

Group the coefficients:

$(\frac{1}{3} + \frac{- 2}{5}) x = (\frac{2}{5} \cdot x) - \frac{2}{5} x$

Find the lowest common denominator:

$(\frac{(1 \cdot 5)}{(3 \cdot 5)} + \frac{(- 2 \cdot 3)}{(5 \cdot 3)}) x = (\frac{2}{5} \cdot x) - \frac{2}{5} x$

Multiply the denominators:

$(\frac{(1 \cdot 5)}{15} + \frac{(- 2 \cdot 3)}{15}) x = (\frac{2}{5} \cdot x) - \frac{2}{5} x$

Multiply the numerators:

$(\frac{5}{15} + \frac{- 6}{15}) x = (\frac{2}{5} \cdot x) - \frac{2}{5} x$

Combine the fractions:

$\frac{(5 - 6)}{15} \cdot x = (\frac{2}{5} \cdot x) - \frac{2}{5} x$

Combine the numerators:

$\frac{- 1}{15} \cdot x = (\frac{2}{5} \cdot x) - \frac{2}{5} x$

Combine the fractions:

$\frac{- 1}{15} \cdot x = \frac{(2 - 2)}{5} x$

Combine the numerators:

$\frac{- 1}{15} \cdot x = \frac{0}{5} x$

Reduce the zero numerator:

$\frac{- 1}{15} x = 0 x$

Simplify the arithmetic:

$\frac{- 1}{15} x = 0$

Divide both sides by the coefficient:

$x = 0$

16 additional steps

$\frac{1}{3} \cdot x = \frac{- 2}{5} x$

Multiply both sides by inverse fraction :

$(\frac{1}{3} x) \cdot \frac{3}{1} = (\frac{- 2}{5} x) \cdot \frac{3}{1}$

Group like terms:

$(\frac{1}{3} \cdot 3) x = (\frac{- 2}{5} x) \cdot \frac{3}{1}$

Multiply the coefficients:

$\frac{(1 \cdot 3)}{3} \cdot x = (\frac{- 2}{5} x) \cdot \frac{3}{1}$

Simplify the fraction:

$x = (\frac{- 2}{5} x) \cdot \frac{3}{1}$

Group like terms:

$x = (\frac{- 2}{5} \cdot 3) x$

Multiply the coefficients:

$x = \frac{(- 2 \cdot 3)}{5} x$

Simplify the arithmetic:

$x = \frac{- 6}{5} x$

Add to both sides:

$x + \frac{6}{5} \cdot x = (\frac{- 6}{5} x) + \frac{6}{5} x$

Group the coefficients:

$(1 + \frac{6}{5}) x = (\frac{- 6}{5} \cdot x) + \frac{6}{5} x$

Convert the integer into a fraction:

$(\frac{5}{5} + \frac{6}{5}) x = (\frac{- 6}{5} \cdot x) + \frac{6}{5} x$

Combine the fractions:

$\frac{(5 + 6)}{5} \cdot x = (\frac{- 6}{5} \cdot x) + \frac{6}{5} x$

Combine the numerators:

$\frac{11}{5} \cdot x = (\frac{- 6}{5} \cdot x) + \frac{6}{5} x$

Combine the fractions:

$\frac{11}{5} \cdot x = \frac{(- 6 + 6)}{5} x$

Combine the numerators:

$\frac{11}{5} \cdot x = \frac{0}{5} x$

Reduce the zero numerator:

$\frac{11}{5} x = 0 x$

Simplify the arithmetic:

$\frac{11}{5} x = 0$

Divide both sides by the coefficient:

$x = 0$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{1}{3} x |$
$y = | \frac{2}{5} x |$
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 x

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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