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Solution - Absolute value equations

Exact form:

m = 9

m=9

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
$| - \frac{2}{3} m + 4 | = | - \frac{2}{3} m + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - \frac{2}{3} m + 4 \| = \| - \frac{2}{3} m + 8 \|$
$x = + y$	$(- \frac{2}{3} m + 4) = (- \frac{2}{3} m + 8)$
$x = - y$	$(- \frac{2}{3} m + 4) = - (- \frac{2}{3} m + 8)$
$+ x = y$	$(- \frac{2}{3} m + 4) = (- \frac{2}{3} m + 8)$
$- x = y$	$- (- \frac{2}{3} m + 4) = (- \frac{2}{3} m + 8)$

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:

$\| x \| = \| y \|$	$\| - \frac{2}{3} m + 4 \| = \| - \frac{2}{3} m + 8 \|$
$x = + y$ , $+ x = y$	$(- \frac{2}{3} m + 4) = (- \frac{2}{3} m + 8)$
$x = - y$ , $- x = y$	$(- \frac{2}{3} m + 4) = - (- \frac{2}{3} m + 8)$

2. Solve the two equations for m

11 additional steps

$(\frac{- 2}{3} \cdot m + 4) = (\frac{- 2}{3} m + 8)$

Add to both sides:

$(\frac{- 2}{3} m + 4) + \frac{2}{3} \cdot m = (\frac{- 2}{3} m + 8) + \frac{2}{3} m$

Group like terms:

$(\frac{- 2}{3} \cdot m + \frac{2}{3} \cdot m) + 4 = (\frac{- 2}{3} \cdot m + 8) + \frac{2}{3} m$

Combine the fractions:

$\frac{(- 2 + 2)}{3} \cdot m + 4 = (\frac{- 2}{3} \cdot m + 8) + \frac{2}{3} m$

Combine the numerators:

$\frac{0}{3} \cdot m + 4 = (\frac{- 2}{3} \cdot m + 8) + \frac{2}{3} m$

Reduce the zero numerator:

$0 m + 4 = (\frac{- 2}{3} \cdot m + 8) + \frac{2}{3} m$

Simplify the arithmetic:

$4 = (\frac{- 2}{3} \cdot m + 8) + \frac{2}{3} m$

Group like terms:

$4 = (\frac{- 2}{3} \cdot m + \frac{2}{3} m) + 8$

Combine the fractions:

$4 = \frac{(- 2 + 2)}{3} m + 8$

Combine the numerators:

$4 = \frac{0}{3} m + 8$

Reduce the zero numerator:

$4 = 0 m + 8$

Simplify the arithmetic:

$4 = 8$

The statement is false:

$4 = 8$

The equation is false so it has no solution.

21 additional steps

$(\frac{- 2}{3} m + 4) = - (\frac{- 2}{3} m + 8)$

Expand the parentheses:

$(\frac{- 2}{3} \cdot m + 4) = \frac{2}{3} m - 8$

Subtract from both sides:

$(\frac{- 2}{3} m + 4) - \frac{2}{3} \cdot m = (\frac{2}{3} m - 8) - \frac{2}{3} m$

Group like terms:

$(\frac{- 2}{3} \cdot m + \frac{- 2}{3} \cdot m) + 4 = (\frac{2}{3} \cdot m - 8) - \frac{2}{3} m$

Combine the fractions:

$\frac{(- 2 - 2)}{3} \cdot m + 4 = (\frac{2}{3} \cdot m - 8) - \frac{2}{3} m$

Combine the numerators:

$\frac{- 4}{3} \cdot m + 4 = (\frac{2}{3} \cdot m - 8) - \frac{2}{3} m$

Group like terms:

$\frac{- 4}{3} \cdot m + 4 = (\frac{2}{3} \cdot m + \frac{- 2}{3} m) - 8$

Combine the fractions:

$\frac{- 4}{3} \cdot m + 4 = \frac{(2 - 2)}{3} m - 8$

Combine the numerators:

$\frac{- 4}{3} \cdot m + 4 = \frac{0}{3} m - 8$

Reduce the zero numerator:

$\frac{- 4}{3} m + 4 = 0 m - 8$

Simplify the arithmetic:

$\frac{- 4}{3} m + 4 = - 8$

Subtract from both sides:

$(\frac{- 4}{3} m + 4) - 4 = - 8 - 4$

Simplify the arithmetic:

$\frac{- 4}{3} m = - 8 - 4$

Simplify the arithmetic:

$\frac{- 4}{3} m = - 12$

Multiply both sides by inverse fraction :

$(\frac{- 4}{3} m) \cdot \frac{3}{- 4} = - 12 \cdot \frac{3}{- 4}$

Move the negative sign from the denominator to the numerator:

$\frac{- 4}{3} m \cdot \frac{- 3}{4} = - 12 \cdot \frac{3}{- 4}$

Group like terms:

$(\frac{- 4}{3} \cdot \frac{- 3}{4}) m = - 12 \cdot \frac{3}{- 4}$

Multiply the coefficients:

$\frac{(- 4 \cdot - 3)}{(3 \cdot 4)} m = - 12 \cdot \frac{3}{- 4}$

Simplify the arithmetic:

$1 m = - 12 \cdot \frac{3}{- 4}$

$m = - 12 \cdot \frac{3}{- 4}$

Move the negative sign from the denominator to the numerator:

$m = - 12 \cdot \frac{- 3}{4}$

Multiply the fraction(s):

$m = \frac{(- 12 \cdot - 3)}{4}$

Simplify the arithmetic:

$m = 9$

3. Graph

Each line represents the function of one side of the equation:
$y = | - \frac{2}{3} m + 4 |$
$y = | - \frac{2}{3} m + 8 |$
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 m

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 m

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved