Question - Solving Absolute Value Inequalities
Solution:
To solve the inequality |4q - 1| ≤ 85, you need to consider the definition of the absolute value. The absolute value of a number is the distance of that number from 0 on the number line, which is always nonnegative. Therefore, |x| ≤ a means that x is within the distance of a from 0. This leads to two scenarios: x ≤ a and x ≥ -a.Applying this idea to |4q - 1| ≤ 85, we get two inequalities:1. 4q - 1 ≤ 85 2. 4q - 1 ≥ -85Now, let's solve each inequality separately.For the first inequality:4q - 1 ≤ 85Add 1 to both sides:4q ≤ 86Divide both sides by 4:q ≤ 86 / 4q ≤ 21.5For the second inequality:4q - 1 ≥ -85Add 1 to both sides:4q ≥ -84Divide both sides by 4:q ≥ -84 / 4q ≥ -21Combining these two inequalities gives us the compound inequality:-21 ≤ q ≤ 21.5This is the solution to the original inequality, which means that q must be greater than or equal to -21 and less than or equal to 21.5.
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