Question - Solving Absolute Value Inequalities
Solution:
To solve the inequality |u + 6| ≥ 46, we have to consider the definition of absolute value. The expression |u + 6| represents the distance of u + 6 from zero on the number line, and this distance is greater than or equal to 46. We can break this into two separate cases, one for u + 6 being positive and one for u + 6 being negative.Case 1 (u + 6 is positive or 0):u + 6 ≥ 46u ≥ 46 - 6u ≥ 40Case 2 (u + 6 is negative):-(u + 6) ≥ 46-u - 6 ≥ 46-u ≥ 46 + 6-u ≥ 52Multiply both sides by -1 and reverse the inequality sign (because multiplying by a negative number reverses the inequality):u ≤ -52So the solution to the inequality |u + 6| ≥ 46 is:u ≥ 40 or u ≤ -52
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