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Solution - Absolute value equations

Exact form:

m = 0, \frac{9}{2}

m=0 , \frac{9}{2}

Mixed number form:

m = 0, 4 \frac{1}{2}

m=0 , 4\frac{1}{2}

Decimal form:

m = 0, 4.5

m=0 , 4.5

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
$| m - 9 | = | 3 m - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| m - 9 \| = \| 3 m - 9 \|$
$x = + y$	$(m - 9) = (3 m - 9)$
$x = - y$	$(m - 9) = - (3 m - 9)$
$+ x = y$	$(m - 9) = (3 m - 9)$
$- x = y$	$- (m - 9) = (3 m - 9)$

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:

$\| x \| = \| y \|$	$\| m - 9 \| = \| 3 m - 9 \|$
$x = + y$ , $+ x = y$	$(m - 9) = (3 m - 9)$
$x = - y$ , $- x = y$	$(m - 9) = - (3 m - 9)$

2. Solve the two equations for m

8 additional steps

$(m - 9) = (3 m - 9)$

Subtract from both sides:

$(m - 9) - 3 m = (3 m - 9) - 3 m$

Group like terms:

$(m - 3 m) - 9 = (3 m - 9) - 3 m$

Simplify the arithmetic:

$- 2 m - 9 = (3 m - 9) - 3 m$

Group like terms:

$- 2 m - 9 = (3 m - 3 m) - 9$

Simplify the arithmetic:

$- 2 m - 9 = - 9$

Add to both sides:

$(- 2 m - 9) + 9 = - 9 + 9$

Simplify the arithmetic:

$- 2 m = - 9 + 9$

Simplify the arithmetic:

$- 2 m = 0$

Divide both sides by the coefficient:

$m = 0$

12 additional steps

$(m - 9) = - (3 m - 9)$

Expand the parentheses:

$(m - 9) = - 3 m + 9$

Add to both sides:

$(m - 9) + 3 m = (- 3 m + 9) + 3 m$

Group like terms:

$(m + 3 m) - 9 = (- 3 m + 9) + 3 m$

Simplify the arithmetic:

$4 m - 9 = (- 3 m + 9) + 3 m$

Group like terms:

$4 m - 9 = (- 3 m + 3 m) + 9$

Simplify the arithmetic:

$4 m - 9 = 9$

Add to both sides:

$(4 m - 9) + 9 = 9 + 9$

Simplify the arithmetic:

$4 m = 9 + 9$

Simplify the arithmetic:

$4 m = 18$

Divide both sides by :

$\frac{(4 m)}{4} = \frac{18}{4}$

Simplify the fraction:

$m = \frac{18}{4}$

Find the greatest common factor of the numerator and denominator:

$m = \frac{(9 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$m = \frac{9}{2}$

3. List the solutions

$m = 0, \frac{9}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | m - 9 |$
$y = | 3 m - 9 |$
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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