Numbers and Relations: Sets, Intervals, Absolute Value, Powers and Expressions
What you must understand
This domain tests fluency with the real number system and with algebraic manipulation. Be comfortable with: subsets of the reals (natural, integer, rational, irrational), operations on sets (union, intersection, difference, complement), interval notation and its mixing with inequalities, absolute value as distance, laws of exponents and roots, and simplification of algebraic expressions including rational expressions and formulas with variables.
Traps that appear in questions
- Endpoint inclusion. A round bracket excludes, a square bracket includes. A question asking for integers in an interval like ((-2, 3\rangle) wants (-1, 0, 1, 2, 3) - not (-2).
- Absolute value equations. (|x-a|=b) has two solutions when (b>0), one when (b=0), none when (b<0). Forgetting the second branch is the classic error.
- Inequalities with negative multipliers. Multiplying or dividing by a negative number flips the inequality sign.
- Rational vs irrational. (\sqrt{9}) is rational; (\sqrt{2}) is not. (0.\overline{3}) is rational.
- Domain restrictions. A denominator may not be zero; an even root needs a non-negative radicand.
Worked example
Solve (|2x-5|\le 7). Rewrite as a double inequality: (-7 \le 2x-5 \le 7). Add 5: (-2 \le 2x \le 12). Divide by 2 (positive, so signs stay): (-1 \le x \le 6). The solution set is the closed interval (\langle -1, 6\rangle).
Rule of thumb: whenever you see an absolute value, an even root, or a fraction with a variable denominator, immediately ask "what values are forbidden or doubled?" before computing.