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Solution - Absolute value equations

Exact form:

r = 2, 1

r=2 , 1

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

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

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:

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

2. Solve the two equations for r

9 additional steps

$r = (3 r - 4)$

Subtract from both sides:

$r - 3 r = (3 r - 4) - 3 r$

Simplify the arithmetic:

$- 2 r = (3 r - 4) - 3 r$

Group like terms:

$- 2 r = (3 r - 3 r) - 4$

Simplify the arithmetic:

$- 2 r = - 4$

Divide both sides by :

$\frac{(- 2 r)}{- 2} = \frac{- 4}{- 2}$

Cancel out the negatives:

$\frac{2 r}{2} = \frac{- 4}{- 2}$

Simplify the fraction:

$r = \frac{- 4}{- 2}$

Cancel out the negatives:

$r = \frac{4}{2}$

Find the greatest common factor of the numerator and denominator:

$r = \frac{(2 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$r = 2$

7 additional steps

$r = - (3 r - 4)$

Expand the parentheses:

$r = - 3 r + 4$

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 r = 4$

Divide both sides by :

$\frac{(4 r)}{4} = \frac{4}{4}$

Simplify the fraction:

$r = \frac{4}{4}$

Simplify the fraction:

$r = 1$

3. List the solutions

$r = 2, 1$
(2 solution(s))

4. Graph

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

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 r

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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