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Solution - Absolute value equations

Exact form:

m = - 8, 2

m=-8 , 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
$| 2 m + 1 | = | m - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 m + 1 \| = \| m - 7 \|$
$x = + y$	$(2 m + 1) = (m - 7)$
$x = - y$	$(2 m + 1) = - (m - 7)$
$+ x = y$	$(2 m + 1) = (m - 7)$
$- x = y$	$- (2 m + 1) = (m - 7)$

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:

$\| x \| = \| y \|$	$\| 2 m + 1 \| = \| m - 7 \|$
$x = + y$ , $+ x = y$	$(2 m + 1) = (m - 7)$
$x = - y$ , $- x = y$	$(2 m + 1) = - (m - 7)$

2. Solve the two equations for m

7 additional steps

$(2 m + 1) = (m - 7)$

Subtract from both sides:

$(2 m + 1) - m = (m - 7) - m$

Group like terms:

$(2 m - m) + 1 = (m - 7) - m$

Simplify the arithmetic:

$m + 1 = (m - 7) - m$

Group like terms:

$m + 1 = (m - m) - 7$

Simplify the arithmetic:

$m + 1 = - 7$

Subtract from both sides:

$(m + 1) - 1 = - 7 - 1$

Simplify the arithmetic:

$m = - 7 - 1$

Simplify the arithmetic:

$m = - 8$

12 additional steps

$(2 m + 1) = - (m - 7)$

Expand the parentheses:

$(2 m + 1) = - m + 7$

Add to both sides:

$(2 m + 1) + m = (- m + 7) + m$

Group like terms:

$(2 m + m) + 1 = (- m + 7) + m$

Simplify the arithmetic:

$3 m + 1 = (- m + 7) + m$

Group like terms:

$3 m + 1 = (- m + m) + 7$

Simplify the arithmetic:

$3 m + 1 = 7$

Subtract from both sides:

$(3 m + 1) - 1 = 7 - 1$

Simplify the arithmetic:

$3 m = 7 - 1$

Simplify the arithmetic:

$3 m = 6$

Divide both sides by :

$\frac{(3 m)}{3} = \frac{6}{3}$

Simplify the fraction:

$m = \frac{6}{3}$

Find the greatest common factor of the numerator and denominator:

$m = \frac{(2 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$m = 2$

3. List the solutions

$m = - 8, 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 m + 1 |$
$y = | m - 7 |$
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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