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Solution - Absolute value equations

Exact form:

m = 3, \frac{1}{5}

m=3 , \frac{1}{5}

Decimal form:

m = 3, 0.2

m=3 , 0.2

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
$4 | 2 m + 1 | = 4 | 3 m - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| 2 m + 1 \| = 4 \| 3 m - 2 \|$
$x = + y$	$4 (2 m + 1) = 4 (3 m - 2)$
$x = - y$	$4 (2 m + 1) = 4 (- (3 m - 2))$
$+ x = y$	$4 (2 m + 1) = 4 (3 m - 2)$
$- x = y$	$4 (- (2 m + 1)) = 4 (3 m - 2)$

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:

$\| x \| = \| y \|$	$4 \| 2 m + 1 \| = 4 \| 3 m - 2 \|$
$x = + y$ , $+ x = y$	$4 (2 m + 1) = 4 (3 m - 2)$
$x = - y$ , $- x = y$	$4 (2 m + 1) = 4 (- (3 m - 2))$

2. Solve the two equations for m

19 additional steps

$4 \cdot (2 m + 1) = 4 \cdot (3 m - 2)$

Expand the parentheses:

$4 \cdot 2 m + 4 \cdot 1 = 4 \cdot (3 m - 2)$

Multiply the coefficients:

$8 m + 4 \cdot 1 = 4 \cdot (3 m - 2)$

Simplify the arithmetic:

$8 m + 4 = 4 \cdot (3 m - 2)$

Expand the parentheses:

$8 m + 4 = 4 \cdot 3 m + 4 \cdot - 2$

Multiply the coefficients:

$8 m + 4 = 12 m + 4 \cdot - 2$

Simplify the arithmetic:

$8 m + 4 = 12 m - 8$

Subtract from both sides:

$(8 m + 4) - 12 m = (12 m - 8) - 12 m$

Group like terms:

$(8 m - 12 m) + 4 = (12 m - 8) - 12 m$

Simplify the arithmetic:

$- 4 m + 4 = (12 m - 8) - 12 m$

Group like terms:

$- 4 m + 4 = (12 m - 12 m) - 8$

Simplify the arithmetic:

$- 4 m + 4 = - 8$

Subtract from both sides:

$(- 4 m + 4) - 4 = - 8 - 4$

Simplify the arithmetic:

$- 4 m = - 8 - 4$

Simplify the arithmetic:

$- 4 m = - 12$

Divide both sides by :

$\frac{(- 4 m)}{- 4} = \frac{- 12}{- 4}$

Cancel out the negatives:

$\frac{4 m}{4} = \frac{- 12}{- 4}$

Simplify the fraction:

$m = \frac{- 12}{- 4}$

Cancel out the negatives:

$m = \frac{12}{4}$

Find the greatest common factor of the numerator and denominator:

$m = \frac{(3 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$m = 3$

18 additional steps

$4 \cdot (2 m + 1) = 4 \cdot (- (3 m - 2))$

Expand the parentheses:

$4 \cdot 2 m + 4 \cdot 1 = 4 \cdot (- (3 m - 2))$

Multiply the coefficients:

$8 m + 4 \cdot 1 = 4 \cdot (- (3 m - 2))$

Simplify the arithmetic:

$8 m + 4 = 4 \cdot (- (3 m - 2))$

Expand the parentheses:

$8 m + 4 = 4 \cdot (- 3 m + 2)$

Expand the parentheses:

$8 m + 4 = 4 \cdot - 3 m + 4 \cdot 2$

Multiply the coefficients:

$8 m + 4 = - 12 m + 4 \cdot 2$

Simplify the arithmetic:

$8 m + 4 = - 12 m + 8$

Add to both sides:

$(8 m + 4) + 12 m = (- 12 m + 8) + 12 m$

Group like terms:

$(8 m + 12 m) + 4 = (- 12 m + 8) + 12 m$

Simplify the arithmetic:

$20 m + 4 = (- 12 m + 8) + 12 m$

Group like terms:

$20 m + 4 = (- 12 m + 12 m) + 8$

Simplify the arithmetic:

$20 m + 4 = 8$

Subtract from both sides:

$(20 m + 4) - 4 = 8 - 4$

Simplify the arithmetic:

$20 m = 8 - 4$

Simplify the arithmetic:

$20 m = 4$

Divide both sides by :

$\frac{(20 m)}{20} = \frac{4}{20}$

Simplify the fraction:

$m = \frac{4}{20}$

Find the greatest common factor of the numerator and denominator:

$m = \frac{(1 \cdot 4)}{(5 \cdot 4)}$

Factor out and cancel the greatest common factor:

$m = \frac{1}{5}$

3. List the solutions

$m = 3, \frac{1}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 4 | 2 m + 1 |$
$y = 4 | 3 m - 2 |$
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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