Decision and conditions
Use endpoint, sign and domain constraints to find permitted values. A real interval may have an unattained endpoint; integer counts require checking strictness.
Worked example
How many integers x satisfy both (x−3)(x−8)≤0 and (x−5)(x−10)≤0? The first inequality allows [3,8], and the second allows [5,10]. Both hold on [5,8]. This intersection contains four integers, including its endpoints. Method choice: Find the closed interval allowed by each inequality, then take their intersection.
Contrast: change the reasoning
If 1≤x≤5 and 7≤y≤11, what is the maximum value of (x−y)²? Every allowed y exceeds every allowed x. The greatest separation is (11)−1 = 10. Squaring gives 100. Method choice: Compare the largest possible separation, not the largest two coordinates together.
Transfer with solution
How many integers x with −3≤x≤5 satisfy (x−3)/(x−4)>0? The ratio is positive for x<3 or x>4. The allowed integers include −3 through 2, plus 5. The count is 7. Method choice: Separate the numerator zero from the forbidden denominator zero.
Boundary check
For real x, |x−3|=2x. What is x? If x≥3, the equation x−3=2x gives a negative value, outside that branch. For 0≤x<3, 3−x=2x, so 3x=3. The solution is 1. Method choice: The right side forces x to be nonnegative; check both absolute-value branches.
Misconception to reject
An extremum of a squared difference can come from opposite endpoints. Takes the union of the two allowed intervals instead of their intersection.
How many integers x satisfy both (x−3)(x−8)≤0 and (x−5)(x−10)≤0?
- The first inequality allows [3,8], and the second allows [5,10].
- Both hold on [5,8].
- This intersection contains four integers, including its endpoints.
Compare the other answer choices
Finds the intersection but excludes its two endpoints despite non-strict inequalities.
Subtracts the intersection endpoints without adding one for an inclusive count.
Counts either single interval without imposing the other condition.
Takes the union of the two allowed intervals instead of their intersection.
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.
GMAT exam content