Question - Solving Absolute Value Inequality
Solution:
To solve the inequality $$ 9 - 8|r + 5| > -11 $$, we need to isolate the absolute value expression on one side.First, let's move the constant term on the left to the other side by subtracting 9 from both sides:$$ 9 - 8|r + 5| - 9 > -11 - 9 $$$$ -8|r + 5| > -20 $$Next, to get $$ |r + 5| $$ by itself, divide both sides of the inequality by -8. Remember that dividing by a negative number flips the direction of the inequality:$$ \frac{-8|r + 5|}{-8} < \frac{-20}{-8} $$$$ |r + 5| < \frac{20}{8} $$$$ |r + 5| < \frac{5}{2} $$ or $$ |r + 5| < 2.5 $$The absolute value inequality $$ |r + 5| < 2.5 $$ means that $$ r + 5 $$ must be less than 2.5 and greater than -2.5 (because the distance from zero is less than 2.5). So we can break this into two separate inequalities:$$ r + 5 < 2.5 $$$$ r + 5 > -2.5 $$Let's solve each of these:For $$ r + 5 < 2.5 $$:Subtract 5 from both sides:$$ r < 2.5 - 5 $$$$ r < -2.5 $$For $$ r + 5 > -2.5 $$:Subtract 5 from both sides:$$ r > -2.5 - 5 $$$$ r > -7.5 $$Combining both inequalities, we get the solution for $$ r $$:$$ -7.5 < r < -2.5 $$This is the set of all real numbers greater than -7.5 and less than -2.5.
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