HSPT: Comparisons: test the whole range
Learn to compare a variable with a number when you only know its range, by testing the allowed values at both ends.
How do you compare quantities when a variable can be any of several integers?
Short answer
List the integers the variable can be and test the smallest and largest. If every allowed value gives the same result, choose it; if any two disagree, the relationship cannot be determined.
Know first: Inequality symbols, Integers, including negatives
Step 1 of 5
The answer must hold for every allowed value
Some comparisons use a variable with a range, such as x is an integer with 10 ≤ x ≤ 15. Here x could be any of several numbers, and the answer you choose has to be true for every one of them.
If one allowed value makes A greater and another makes B greater, or one makes them equal, the answer is the relationship cannot be determined.
| Symbol | Meaning | Integers allowed when the range is 8 to 10 |
|---|---|---|
| 8 ≤ x ≤ 10 | 8 and 10 are included | 8, 9, 10 |
| 8 < x < 10 | 8 and 10 are left out | 9 only |
| 8 ≤ x < 10 | 8 is in, 10 is out | 8, 9 |
Integer means a whole number, which can be negative or zero.
Step 2 of 5
Try the smallest and largest values
Worked example: A range that crosses the number
x is an integer with 10 ≤ x ≤ 15. Compare Quantity A: x and Quantity B: 12.
- AQuantity A is greater.
- BQuantity B is greater.
- CThe two quantities are equal.
- DThe relationship cannot be determined.
- 1
List the values
x can be 10, 11, 12, 13, 14 or 15.
- 2
Test the smallest
x = 10: 10 is less than 12, so B is greater.
- 3
Test the largest
x = 15: 15 is greater than 12, so A is greater.
- 4
Decide
The ends disagree, and x = 12 even makes them equal. No single answer holds every time.
- 5
See why the other choices are there
A, B and equal are each true for some values of x, which is why they tempt you. None is true for all of them.
Answer
The relationship cannot be determined.
Wrong: 8 < x < 10 allows 8, 9 and 10.
Right: 8 < x < 10 allows only 9.
The symbol < leaves the endpoint out; only ≤ includes it. Misreading the symbol changes which values you test.
Step 3 of 5
Check every allowed value
Check yourself · Question 1
x is an integer with 8 < x < 10. Compare Quantity A: x and Quantity B: 9.
Answer: C
The only integer greater than 8 and less than 10 is 9. So x must be 9, and the two quantities are equal.
Trap answer: The relationship cannot be determined.
Assuming a variable always varies
- Why it looks right
- A letter usually stands for many values, so cannot be determined feels like the safe choice.
- Why it's wrong
- With strict inequalities, only one integer fits between 8 and 10. Once x is pinned down, the comparison is fixed.
- The right answer
- The two quantities are equal, because x must be 9.
Check yourself · Question 2
x is an integer with 1 ≤ x ≤ 4. Compare Quantity A: 3x and Quantity B: 13.
Answer: C
The largest allowed value is x = 4, which gives 3x = 12. Even at its biggest, A is less than 13, so B is always greater.
Check yourself · Question 3
x is an integer with −2 ≤ x ≤ 2. Compare Quantity A: x² and Quantity B: 4.
Answer: C
The squares of −2, −1, 0, 1 and 2 are 4, 1, 0, 1 and 4. When x = 2, the quantities are equal; when x = 0, B is greater. They disagree, so the relationship cannot be determined.
Common mistake
"I tried one value and it worked, so that's the answer."
- Why it's tempting
- One quick test feels like proof, especially when time is short.
- Do this instead
- One value can only show what is possible. Test the smallest and largest allowed values, and watch for any value that gives a different result.
Step 4 of 5
What to remember
Remember
- A comparison answer must hold for every allowed value of the variable.
- Read the symbols: ≤ includes the endpoint, < leaves it out.
- If two allowed values give different results, the relationship cannot be determined.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
Which values of x did you test in the practice question, and did every one give the same result?