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Solution - Absolute value equations

Exact form:

m = - 7, \frac{7}{3}

m=-7 , \frac{7}{3}

Mixed number form:

m = - 7, 2 \frac{1}{3}

m=-7 , 2\frac{1}{3}

Decimal form:

m = - 7, 2.333

m=-7 , 2.333

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
$| 2 m | = | m - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 m \| = \| m - 7 \|$
$x = + y$	$(2 m) = (m - 7)$
$x = - y$	$(2 m) = - (m - 7)$
$+ x = y$	$(2 m) = (m - 7)$
$- x = y$	$- (2 m) = (m - 7)$

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:

$\| x \| = \| y \|$	$\| 2 m \| = \| m - 7 \|$
$x = + y$ , $+ x = y$	$(2 m) = (m - 7)$
$x = - y$ , $- x = y$	$(2 m) = - (m - 7)$

2. Solve the two equations for m

3 additional steps

$2 m = (m - 7)$

Subtract from both sides:

$(2 m) - m = (m - 7) - m$

Simplify the arithmetic:

$m = (m - 7) - m$

Group like terms:

$m = (m - m) - 7$

Simplify the arithmetic:

$m = - 7$

6 additional steps

$2 m = - (m - 7)$

Expand the parentheses:

$2 m = - m + 7$

Add to both sides:

$(2 m) + m = (- m + 7) + m$

Simplify the arithmetic:

$3 m = (- m + 7) + m$

Group like terms:

$3 m = (- m + m) + 7$

Simplify the arithmetic:

$3 m = 7$

Divide both sides by :

$\frac{(3 m)}{3} = \frac{7}{3}$

Simplify the fraction:

$m = \frac{7}{3}$

3. List the solutions

$m = - 7, \frac{7}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 m |$
$y = | m - 7 |$
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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