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Solution - Absolute value equations

Exact form:

m = \frac{2}{3}

m=\frac{2}{3}

Decimal form:

m = 0.667

m=0.667

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

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

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:

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

2. Solve the two equations for m

4 additional steps

$(3 m - 4) = 3 m$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$- 4 = 0$

The statement is false:

$- 4 = 0$

The equation is false so it has no solution.

9 additional steps

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

Add to both sides:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

$6 m = ((- 3 m) + 4) + 3 m$

Group like terms:

$6 m = (- 3 m + 3 m) + 4$

Simplify the arithmetic:

$6 m = 4$

Divide both sides by :

$\frac{(6 m)}{6} = \frac{4}{6}$

Simplify the fraction:

$m = \frac{4}{6}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$m = \frac{2}{3}$

3. Graph

Each line represents the function of one side of the equation:
$y = | 3 m - 4 |$
$y = | 3 m |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved