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Solution - Absolute value equations

Exact form:

m = \frac{3}{2}

m=\frac{3}{2}

Mixed number form:

m = 1 \frac{1}{2}

m=1\frac{1}{2}

Decimal form:

m = 1.5

m=1.5

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

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

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:

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

2. Solve the two equations for m

7 additional steps

$- m = (m - 3)$

Subtract from both sides:

$- m - m = (m - 3) - m$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 m = - 3$

Divide both sides by :

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

Cancel out the negatives:

$\frac{2 m}{2} = \frac{- 3}{- 2}$

Simplify the fraction:

$m = \frac{- 3}{- 2}$

Cancel out the negatives:

$m = \frac{3}{2}$

5 additional steps

$- m = - (m - 3)$

Expand the parentheses:

$- m = - m + 3$

Add to both sides:

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

Simplify the arithmetic:

$0 = (- m + 3) + m$

Group like terms:

$0 = (- m + m) + 3$

Simplify the arithmetic:

$0 = 3$

The statement is false:

$0 = 3$

The equation is false so it has no solution.

3. List the solutions

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

4. Graph

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