Question - Solving Absolute Value Inequalities
Solution:
To solve the absolute value inequality $$ |4q - 1| \leq 85 $$, we must consider the two cases of the absolute value definition:1. The expression inside the absolute value, $$ 4q - 1 $$, is non-negative.2. The expression inside the absolute value, $$ 4q - 1 $$, is negative.For the first case, when $$ 4q - 1 $$ is non-negative:\[ 4q - 1 \leq 85 \]To solve this inequality, add 1 to both sides:\[ 4q \leq 86 \]Now, divide both sides by 4:\[ q \leq 21.5 \]For the second case, when $$ 4q - 1 $$ is negative, we consider the opposite of the expression, which gives us:\[ -(4q - 1) \leq 85 \]Simplify the left side by distributing the negative sign:\[ -4q + 1 \leq 85 \]Now, to isolate the q term, subtract 1 from both sides:\[ -4q \leq 84 \]Divide both sides by -4 and remember to reverse the inequality sign because we are dividing by a negative number:\[ q \geq -21 \]Putting both cases together gives us the compound inequality:\[ -21 \leq q \leq 21.5 \]This is the solution set for the inequality $$ |4q - 1| \leq 85 $$.
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