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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

m = 1, 0.333

m=1 , 0.333

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| - 3 m + 1 | + | 3 m - 1 | = 0$

Add $- | 3 m - 1 |$ to both sides of the equation:

$| - 3 m + 1 | + | 3 m - 1 | - | 3 m - 1 | = - | 3 m - 1 |$

Simplify the arithmetic

$| - 3 m + 1 | = - | 3 m - 1 |$

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

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

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:

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

3. Solve the two equations for m

5 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$1 = 1$

14 additional steps

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

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 6 m + 1 = (3 m - 1) - 3 m$

Group like terms:

$- 6 m + 1 = (3 m - 3 m) - 1$

Simplify the arithmetic:

$- 6 m + 1 = - 1$

Subtract from both sides:

$(- 6 m + 1) - 1 = - 1 - 1$

Simplify the arithmetic:

$- 6 m = - 1 - 1$

Simplify the arithmetic:

$- 6 m = - 2$

Divide both sides by :

$\frac{(- 6 m)}{- 6} = \frac{- 2}{- 6}$

Cancel out the negatives:

$\frac{6 m}{6} = \frac{- 2}{- 6}$

Simplify the fraction:

$m = \frac{- 2}{- 6}$

Cancel out the negatives:

$m = \frac{2}{6}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$m = \frac{1}{3}$

4. List the solutions

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

5. Graph

Each line represents the function of one side of the equation:
$y = | - 3 m + 1 |$
$y = - | 3 m - 1 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for m

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for m

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved