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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
$\frac{11}{3} | x | = \frac{2}{3} | x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\frac{11}{3} \| x \| = \frac{2}{3} \| x \|$
$x = + y$	$\frac{11}{3} (x) = \frac{2}{3} (x)$
$x = - y$	$\frac{11}{3} (x) = \frac{2}{3} (- (x))$
$+ x = y$	$\frac{11}{3} (x) = \frac{2}{3} (x)$
$- x = y$	$\frac{11}{3} (- (x)) = \frac{2}{3} (x)$

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:

$\| x \| = \| y \|$	$\frac{11}{3} \| x \| = \frac{2}{3} \| x \|$
$x = + y$ , $+ x = y$	$\frac{11}{3} (x) = \frac{2}{3} (x)$
$x = - y$ , $- x = y$	$\frac{11}{3} (x) = \frac{2}{3} (- (x))$

2. Solve the two equations for x

9 additional steps

$\frac{11}{3} \cdot x = \frac{2}{3} x$

Subtract from both sides:

$(\frac{11}{3} x) - \frac{2}{3} \cdot x = (\frac{2}{3} x) - \frac{2}{3} x$

Combine the fractions:

$\frac{(11 - 2)}{3} \cdot x = (\frac{2}{3} \cdot x) - \frac{2}{3} x$

Combine the numerators:

$\frac{9}{3} \cdot x = (\frac{2}{3} \cdot x) - \frac{2}{3} x$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

Combine the fractions:

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

Combine the numerators:

$3 x = \frac{0}{3} x$

Reduce the zero numerator:

$3 x = 0 x$

Simplify the arithmetic:

$3 x = 0$

Divide both sides by the coefficient:

$x = 0$

10 additional steps

$\frac{11}{3} x = \frac{2}{3} \cdot - x$

Group like terms:

$\frac{11}{3} x = (\frac{2}{3} \cdot - 1) x$

Multiply the coefficients:

$\frac{11}{3} \cdot x = \frac{(2 \cdot - 1)}{3} x$

Simplify the arithmetic:

$\frac{11}{3} \cdot x = \frac{- 2}{3} x$

Add to both sides:

$(\frac{11}{3} x) + \frac{2}{3} \cdot x = (\frac{- 2}{3} x) + \frac{2}{3} x$

Combine the fractions:

$\frac{(11 + 2)}{3} \cdot x = (\frac{- 2}{3} \cdot x) + \frac{2}{3} x$

Combine the numerators:

$\frac{13}{3} \cdot x = (\frac{- 2}{3} \cdot x) + \frac{2}{3} x$

Combine the fractions:

$\frac{13}{3} \cdot x = \frac{(- 2 + 2)}{3} x$

Combine the numerators:

$\frac{13}{3} \cdot x = \frac{0}{3} x$

Reduce the zero numerator:

$\frac{13}{3} x = 0 x$

Simplify the arithmetic:

$\frac{13}{3} x = 0$

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:
$y = \frac{11}{3} | x |$
$y = \frac{2}{3} | x |$
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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