Question - Solving Absolute Value Inequalities
Solution:
To solve the inequality $$ |8r| + 2 > 42 $$, we will first isolate the absolute value part by subtracting 2 from both sides of the inequality:\[ |8r| + 2 - 2 > 42 - 2 \]\[ |8r| > 40 \]Now, by the definition of absolute value, $$ |8r| > 40 $$ means that 8r is either greater than 40 or less than -40. We can split this into two separate inequalities to represent the compound inequality:1) $$ 8r > 40 $$2) $$ 8r < -40 $$Next, we can solve each inequality for r. Starting with the first inequality:\[ 8r > 40 \]\[ r > \frac{40}{8} \]\[ r > 5 \]For the second inequality:\[ 8r < -40 \]\[ r < \frac{-40}{8} \]\[ r < -5 \]Now, we present these two solutions as a compound inequality:\[ r < -5 \text{ or } r > 5 \]This is the solution to the original inequality $$ |8r| + 2 > 42 $$.
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