Solve for r: 3|3 - 5r| - 3 = 18
Understand the Problem
The question asks us to solve an absolute value equation for the variable 'r'. We need to isolate the absolute value term and then consider both positive and negative cases for the expression inside the absolute value.
Answer
$r = -\frac{4}{5}, 2$
Answer for screen readers
$r = -\frac{4}{5}, 2$
Steps to Solve
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Isolate the absolute value term Add 3 to both sides of the equation: $$3|3 - 5r| - 3 + 3 = 18 + 3$$ $$3|3 - 5r| = 21$$
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Divide to isolate the absolute value Divide both sides by 3: $$\frac{3|3 - 5r|}{3} = \frac{21}{3}$$ $$|3 - 5r| = 7$$
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Consider the positive case Set the expression inside the absolute value equal to 7: $$3 - 5r = 7$$
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Solve for r in the positive case Subtract 3 from both sides: $$3 - 5r - 3 = 7 - 3$$ $$-5r = 4$$ Divide by -5: $$r = -\frac{4}{5}$$
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Consider the negative case Set the expression inside the absolute value equal to -7: $$3 - 5r = -7$$
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Solve for r in the negative case Subtract 3 from both sides: $$3 - 5r - 3 = -7 - 3$$ $$-5r = -10$$ Divide by -5: $$r = \frac{-10}{-5} = 2$$
$r = -\frac{4}{5}, 2$
More Information
The absolute value equation has two solutions for $r$: $-\frac{4}{5}$ and $2$.
Tips
A common mistake is forgetting to consider both the positive and negative cases when solving absolute value equations. Another common mistake is incorrectly isolating the absolute value term.
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