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Solution - Absolute value equations

Exact form:

m = 0, 0

m=0 , 0

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

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

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:

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

2. Solve the two equations for m

4 additional steps

$7 m = 2 \cdot 2 m$

Multiply the coefficients:

$7 m = 4 m$

Subtract from both sides:

$(7 m) - 4 m = (4 m) - 4 m$

Simplify the arithmetic:

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

Simplify the arithmetic:

$3 m = 0$

Divide both sides by the coefficient:

$m = 0$

4 additional steps

$7 m = 2 \cdot - 2 m$

Multiply the coefficients:

$7 m = - 4 m$

Add to both sides:

$(7 m) + 4 m = (- 4 m) + 4 m$

Simplify the arithmetic:

$11 m = (- 4 m) + 4 m$

Simplify the arithmetic:

$11 m = 0$

Divide both sides by the coefficient:

$m = 0$

3. List the solutions

$m = 0, 0$
(2 solution(s))

4. Graph

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