SAT Integer Constraints and "Which Could Be" Problems: Testing Integers and Finding Valid Solutions

Some SAT problems state or imply that variables must be integers. Example: "x is an integer and x^2<25. Which could be a value of x?" The constraint is crucial: x could be -4, -3, -2, -1, 0, 1, 2, 3, or 4 (testing which squares are less than 25). Without the integer constraint, x could be any number between -5 and 5 (like 2.5). Integer constraints dramatically limit possible values, but students often overlook the constraint and treat the variable as continuous, leading to wrong answers. Catching the constraint is half the battle.

When you see a "which could be" or "which must be" question, check: is there a constraint? Is the variable an integer, non-negative, whole number, etc.? Mark the constraint. Then systematically test integer values to find which satisfy the constraint. Most students skip systematic testing and guess, getting the question wrong. Testing takes 30 seconds and ensures accuracy.

Step 1: Identify the constraint(s) on x (integer, positive, x<5, etc.). Step 2: List all integers satisfying the constraint. If x is a positive integer less than 5: x ∈ {1, 2, 3, 4}. Step 3: Test each value in the condition. Example: "Which could be a value of x^2+x?" Test x=1: 1+1=2. x=2: 4+2=6. x=3: 9+3=12. x=4: 16+4=20. Step 4: Check which of these values appears in the answer choices. The values you calculated (2, 6, 12, 20) should appear; any answer choice not in this list is wrong. This systematic approach guarantees you find the right answer.

Practice this technique on 8 "which could be" integer-constraint problems. First four: write out all steps. Next four: use steps mentally. By problem 8, the process is automatic. Most students never systematize and thus struggle. You will be able to solve these questions under 45 seconds once the process is automatic.

Example 1: "x is a positive integer and 2x+3<15. Which of the following could be x?" Constraint: positive integer, 2x+3<15 means 2x<12 means x<6. Valid values: x ∈ {1, 2, 3, 4, 5}. If answer choices are {4, 6, 7, 8}, only 4 is valid. Students who do not list valid values might guess 6, 7, or 8 (wrong). Those who systematically test identify 4.

Example 2: "n is a non-negative integer and n^2≤20. Which could be n^2?" Constraint: non-negative, n^2≤20. Valid n: {0, 1, 2, 3, 4} (since 4^2=16, 5^2=25>20). Valid n^2 values: {0, 1, 4, 9, 16}. If answer choices are {10, 16, 25, 30}, only 16 is valid. Without systematic testing, students might guess 25 (wrong).

Complete 12 integer-constraint "which could be" or "which must be" problems over three days (4 per day). Days 1-2: write out steps (list valid integers, test each, identify matches). Day 3: mental steps, no writing except final answer. Track accuracy: aim for 11/12 correct. If accuracy drops below 11, slow down and write out steps. Most mistakes come from incomplete testing (forgetting to check all integers) or arithmetic errors (miscalculating which integers satisfy the constraint). Once you hit 12/12, you are ready for test day. Integer-constraint questions will feel straightforward because you have a system.

On test day, systematic testing on these problems guarantees you will get them right, spending only 45 seconds per question. Speed comes from automating the process, not from rushing.