Numeric Entry & Multiple-Answer • Topic 3 of 4

Checking Every Option

Because a Multiple-Answer question can have any number of correct choices and gives no partial credit, the only safe method is to evaluate every option independently — never stop at the first one that works, and never assume the correct answers cluster together. Treat each choice as its own true/false test against the condition, and mark it only if it genuinely qualifies. For questions phrased as a range ("which could be the value of..."), a smart shortcut is to find the smallest and largest values that satisfy the condition; every option between those two extremes then qualifies, and you can sweep them in one pass. Guard against two habits from ordinary multiple choice: the instinct to pick exactly one answer, and the instinct to pick a "best" answer — here there is no best, only every-correct-one. When the algebra sets a boundary (say x > 4), test the endpoint carefully to decide whether it is included, since a strict versus inclusive inequality changes which options make the cut.

✅ Solved examples

1. Select-all. If x is an integer and 3x < 20, which of these could be x? (A) 4 (B) 6 (C) 7 (D) 10
3x < 20 means x < 6.67. Test each: 4 ✓ (12 < 20), 6 ✓ (18 < 20), 7 ✗ (21 > 20), 10 ✗. Correct: A and B. Checking each prevents assuming 7 sneaks in.
2. Select-all. A number added to 5 gives a result greater than 8 and at most 12. Which could the number be? (A) 2 (B) 4 (C) 7 (D) 8
Condition: 8 < n + 5 ≤ 12, so 3 < n ≤ 7. Test: 2 ✗, 4 ✓, 7 ✓ (endpoint included), 8 ✗. Correct: B and C.
3. Select-all. Which of these could be the length of the third side of a triangle with sides 6 and 10? (A) 3 (B) 5 (C) 12 (D) 16
The third side lies strictly between 10 − 6 = 4 and 10 + 6 = 16, i.e. 4 < side < 16. Test: 3 ✗, 5 ✓, 12 ✓, 16 ✗ (must be strictly less than 16). Correct: B and C.
4. Select-all. If 0 < x < 1, which of the following must be true? (A) x² < x (B) 1/x > 1 (C) x³ > x (D) √x > x
For a proper fraction: x² < x ✓; 1/x > 1 ✓; x³ > x is false (x³ < x) ✗; √x > x ✓. Correct: A, B and D. Each option is a separate must-be-true test.

✏️ Practice — try these, take hints as needed

1. Select-all. If x is an integer and 2x < 15, which could x be? (A) 5 (B) 7 (C) 8 (D) 10
Solve: x < 7.5.
Test each option against 7.5.
8 and 10 exceed it.
A, B
2. Select-all. A number n satisfies 4 ≤ n + 1 < 9. Which could n be? (A) 3 (B) 5 (C) 7 (D) 8
Subtract 1: 3 ≤ n < 8.
3 is included, 8 is not.
Sweep the range.
A, B, C
3. Select-all. Third side of a triangle with sides 5 and 9? (A) 3 (B) 4 (C) 10 (D) 14
Range is 9 − 5 < side < 9 + 5.
That is 4 < side < 14.
Endpoints are excluded.
C
4. Select-all. If −2 < x < 3, which must be true? (A) x² < 9 (B) x > 0 (C) x + 2 > 0 (D) x is an integer
Test each independently.
x could be negative or non-integer.
Check the boundary cases.
A, C
5. Select-all. Which integers n satisfy n² ≤ 10? (A) −3 (B) −2 (C) 0 (D) 3 (E) 4
n² ≤ 10 means |n| ≤ 3.16.
Include negatives.
16 > 10 rules out 4.
A, B, C, D

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Select-One vs Select-All On-Screen Calculator & Estimation →