Decision and conditions
Locate critical points, then test signs on the intervals they create. Zeros can be included only for non-strict inequalities; denominator zeros are always excluded.
Worked model
How many integers x satisfy (x − 3)(x − 8) ≤ 0? The roots are 3 and 8. The solution interval is [3, 8]. Including both endpoints gives six integers. Before calculating: The upward quadratic is nonpositive between its roots; check whether endpoints count.
Contrast: a different decision
Select all listed x satisfying x² > 9. Choices: -4; -3; 0; 3; 4. Equivalent to |x| > 3. Thus x < −3 or x > 3. The listed solutions are -4, 4. Before calculating: A square exceeds the threshold on two outer intervals, not just the positive side.
Transfer with solution
Select all listed x satisfying (x − 3)/(x − 4) > 0. Choices: -3; 0; 3; 4; 5. The numerator is zero at 3; the denominator is zero at 4. The ratio is positive for x < 3 or x > 4. The allowed choices are -3, 0, 5. Before calculating: Mark the numerator zero and forbidden denominator zero before testing signs.
Check the boundary
Multiplying by an expression of unknown sign can reverse an inequality and discard cases. How many integers x with −7 ≤ x ≤ 7 satisfy |x| > 3? The positive solutions are 4 through 7, four values. There are four corresponding negative solutions. The total is eight.
How many integers x satisfy (x − 3)(x − 8) ≤ 0?
- The roots are 3 and 8.
- The solution interval is [3, 8].
- Including both endpoints gives six integers.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overview