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Solution - Absolute value equations

Exact form:

r = \frac{20}{3}, 2

r=\frac{20}{3} , 2

Mixed number form:

r = 6 \frac{2}{3}, 2

r=6\frac{2}{3} , 2

Decimal form:

r = 6.667, 2

r=6.667 , 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
$| 2 r + 3 | = | 5 r - 17 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 r + 3 \| = \| 5 r - 17 \|$
$x = + y$	$(2 r + 3) = (5 r - 17)$
$x = - y$	$(2 r + 3) = - (5 r - 17)$
$+ x = y$	$(2 r + 3) = (5 r - 17)$
$- x = y$	$- (2 r + 3) = (5 r - 17)$

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:

$\| x \| = \| y \|$	$\| 2 r + 3 \| = \| 5 r - 17 \|$
$x = + y$ , $+ x = y$	$(2 r + 3) = (5 r - 17)$
$x = - y$ , $- x = y$	$(2 r + 3) = - (5 r - 17)$

2. Solve the two equations for r

11 additional steps

$(2 r + 3) = (5 r - 17)$

Subtract from both sides:

$(2 r + 3) - 5 r = (5 r - 17) - 5 r$

Group like terms:

$(2 r - 5 r) + 3 = (5 r - 17) - 5 r$

Simplify the arithmetic:

$- 3 r + 3 = (5 r - 17) - 5 r$

Group like terms:

$- 3 r + 3 = (5 r - 5 r) - 17$

Simplify the arithmetic:

$- 3 r + 3 = - 17$

Subtract from both sides:

$(- 3 r + 3) - 3 = - 17 - 3$

Simplify the arithmetic:

$- 3 r = - 17 - 3$

Simplify the arithmetic:

$- 3 r = - 20$

Divide both sides by :

$\frac{(- 3 r)}{- 3} = \frac{- 20}{- 3}$

Cancel out the negatives:

$\frac{3 r}{3} = \frac{- 20}{- 3}$

Simplify the fraction:

$r = \frac{- 20}{- 3}$

Cancel out the negatives:

$r = \frac{20}{3}$

12 additional steps

$(2 r + 3) = - (5 r - 17)$

Expand the parentheses:

$(2 r + 3) = - 5 r + 17$

Add to both sides:

$(2 r + 3) + 5 r = (- 5 r + 17) + 5 r$

Group like terms:

$(2 r + 5 r) + 3 = (- 5 r + 17) + 5 r$

Simplify the arithmetic:

$7 r + 3 = (- 5 r + 17) + 5 r$

Group like terms:

$7 r + 3 = (- 5 r + 5 r) + 17$

Simplify the arithmetic:

$7 r + 3 = 17$

Subtract from both sides:

$(7 r + 3) - 3 = 17 - 3$

Simplify the arithmetic:

$7 r = 17 - 3$

Simplify the arithmetic:

$7 r = 14$

Divide both sides by :

$\frac{(7 r)}{7} = \frac{14}{7}$

Simplify the fraction:

$r = \frac{14}{7}$

Find the greatest common factor of the numerator and denominator:

$r = \frac{(2 \cdot 7)}{(1 \cdot 7)}$

Factor out and cancel the greatest common factor:

$r = 2$

3. List the solutions

$r = \frac{20}{3}, 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 r + 3 |$
$y = | 5 r - 17 |$
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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