Inequalities and Absolute Values
GRE inequality problems at the advanced level require careful attention to the rules governing inequality manipulation. Key rules: you can add or subtract any value from both sides without changing the direction of the inequality; you can multiply or divide by a positive number without changing direction; multiplying or dividing by a negative number reverses the inequality direction. Absolute value inequalities generate two solution cases: |x| < a means −a < x < a (a bounded interval); |x| > a means x < −a or x > a (two unbounded intervals). Common GRE trap: solving |x − 3| > 5 by writing only x > 8 and omitting x < −2. Always write both cases. Inequalities with variables and constraints: if the problem states x > 0, this eliminates negative number substitutions — simplify within that constraint. Systems of inequalities (e.g., x > 2 and y < 5 and x + y < 10) define a feasible region — GRE questions may ask for the maximum or minimum of a linear expression over this region, which occurs at a corner point of the feasible region (the intersection of two constraint boundaries). Compound inequalities: 3 < 2x + 1 < 9 is solved by applying the same operations to all three parts simultaneously: subtract 1 from all three (2 < 2x < 8), divide by 2 (1 < x < 4). The final answer is a range, not a single value.