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Solution - Absolute value equations

Exact form:

m = - 3, - \frac{17}{3}

m=-3 , -\frac{17}{3}

Mixed number form:

m = - 3, - 5 \frac{2}{3}

m=-3 , -5\frac{2}{3}

Decimal form:

m = - 3, - 5.667

m=-3 , -5.667

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
$| m + 7 | = 2 | m + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| m + 7 \| = 2 \| m + 5 \|$
$x = + y$	$(m + 7) = 2 (m + 5)$
$x = - y$	$(m + 7) = 2 (- (m + 5))$
$+ x = y$	$(m + 7) = 2 (m + 5)$
$- x = y$	$- (m + 7) = 2 (m + 5)$

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:

$\| x \| = \| y \|$	$\| m + 7 \| = 2 \| m + 5 \|$
$x = + y$ , $+ x = y$	$(m + 7) = 2 (m + 5)$
$x = - y$ , $- x = y$	$(m + 7) = 2 (- (m + 5))$

2. Solve the two equations for m

12 additional steps

$(m + 7) = 2 \cdot (m + 5)$

Expand the parentheses:

$(m + 7) = 2 m + 2 \cdot 5$

Simplify the arithmetic:

$(m + 7) = 2 m + 10$

Subtract from both sides:

$(m + 7) - 2 m = (2 m + 10) - 2 m$

Group like terms:

$(m - 2 m) + 7 = (2 m + 10) - 2 m$

Simplify the arithmetic:

$- m + 7 = (2 m + 10) - 2 m$

Group like terms:

$- m + 7 = (2 m - 2 m) + 10$

Simplify the arithmetic:

$- m + 7 = 10$

Subtract from both sides:

$(- m + 7) - 7 = 10 - 7$

Simplify the arithmetic:

$- m = 10 - 7$

Simplify the arithmetic:

$- m = 3$

Multiply both sides by :

$- m \cdot - 1 = 3 \cdot - 1$

Remove the one(s):

$m = 3 \cdot - 1$

Simplify the arithmetic:

$m = - 3$

14 additional steps

$(m + 7) = 2 \cdot (- (m + 5))$

Expand the parentheses:

$(m + 7) = 2 \cdot (- m - 5)$

$(m + 7) = 2 \cdot - m + 2 \cdot - 5$

Group like terms:

$(m + 7) = (2 \cdot - 1) m + 2 \cdot - 5$

Multiply the coefficients:

$(m + 7) = - 2 m + 2 \cdot - 5$

Simplify the arithmetic:

$(m + 7) = - 2 m - 10$

Add to both sides:

$(m + 7) + 2 m = (- 2 m - 10) + 2 m$

Group like terms:

$(m + 2 m) + 7 = (- 2 m - 10) + 2 m$

Simplify the arithmetic:

$3 m + 7 = (- 2 m - 10) + 2 m$

Group like terms:

$3 m + 7 = (- 2 m + 2 m) - 10$

Simplify the arithmetic:

$3 m + 7 = - 10$

Subtract from both sides:

$(3 m + 7) - 7 = - 10 - 7$

Simplify the arithmetic:

$3 m = - 10 - 7$

Simplify the arithmetic:

$3 m = - 17$

Divide both sides by :

$\frac{(3 m)}{3} = \frac{- 17}{3}$

Simplify the fraction:

$m = \frac{- 17}{3}$

3. List the solutions

$m = - 3, - \frac{17}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | m + 7 |$
$y = 2 | m + 5 |$
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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