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Solution - Absolute value equations

Exact form:

m = 6, 2

m=6 , 2

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
$| m | = 2 | m - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| m \| = 2 \| m - 3 \|$
$x = + y$	$(m) = 2 (m - 3)$
$x = - y$	$(m) = 2 (- (m - 3))$
$+ x = y$	$(m) = 2 (m - 3)$
$- x = y$	$- (m) = 2 (m - 3)$

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:

$\| x \| = \| y \|$	$\| m \| = 2 \| m - 3 \|$
$x = + y$ , $+ x = y$	$(m) = 2 (m - 3)$
$x = - y$ , $- x = y$	$(m) = 2 (- (m - 3))$

2. Solve the two equations for m

8 additional steps

$m = 2 \cdot (m - 3)$

Expand the parentheses:

$m = 2 m + 2 \cdot - 3$

Simplify the arithmetic:

$m = 2 m - 6$

Subtract from both sides:

$m - 2 m = (2 m - 6) - 2 m$

Simplify the arithmetic:

$- m = (2 m - 6) - 2 m$

Group like terms:

$- m = (2 m - 2 m) - 6$

Simplify the arithmetic:

$- m = - 6$

Multiply both sides by :

$- m \cdot - 1 = - 6 \cdot - 1$

Remove the one(s):

$m = - 6 \cdot - 1$

Simplify the arithmetic:

$m = 6$

12 additional steps

$m = 2 \cdot (- (m - 3))$

Expand the parentheses:

$m = 2 \cdot (- m + 3)$

$m = 2 \cdot - m + 2 \cdot 3$

Group like terms:

$m = (2 \cdot - 1) m + 2 \cdot 3$

Multiply the coefficients:

$m = - 2 m + 2 \cdot 3$

Simplify the arithmetic:

$m = - 2 m + 6$

Add to both sides:

$m + 2 m = (- 2 m + 6) + 2 m$

Simplify the arithmetic:

$3 m = (- 2 m + 6) + 2 m$

Group like terms:

$3 m = (- 2 m + 2 m) + 6$

Simplify the arithmetic:

$3 m = 6$

Divide both sides by :

$\frac{(3 m)}{3} = \frac{6}{3}$

Simplify the fraction:

$m = \frac{6}{3}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$m = 2$

3. List the solutions

$m = 6, 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | m |$
$y = 2 | m - 3 |$
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. 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 m

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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