An integer x is chosen uniformly at random from {1, 2, …, 100}. What is the probability that x satisfies the inequality x^2 − 13x ≤ 30?

Introduction / Context:We must count how many integers between 1 and 100 inclusive satisfy a quadratic inequality, then divide by 100 to obtain the probability.

Given Data / Assumptions:

• Uniform choice over {1, …, 100}.
• Inequality: x^2 − 13x ≤ 30.

Concept / Approach:Solve the inequality by finding the roots of the corresponding quadratic equation and using the sign pattern of a parabola opening upward.

Step-by-Step Solution:Solve x^2 − 13x − 30 = 0.Discriminant Δ = 13^2 + 4*30 = 169 + 120 = 289; √Δ = 17.Roots: (13 ± 17)/2 → x = 15 and x = −2.Since the parabola opens upward, the solution set to x^2 − 13x − 30 ≤ 0 is −2 ≤ x ≤ 15.Within {1, …, 100}, valid x are 1 through 15 inclusive → 15 integers.Probability = 15 / 100 = 3/20.

Verification / Alternative check:Quick test: x = 15 yields 225 − 195 = 30 (boundary satisfied); x = 16 yields 256 − 208 = 48 > 30 (violates), confirming the cutoff.

Why Other Options Are Wrong:Fractions like 5/9, 7/9 are too large; 9/50 = 18% undercounts; 2/5 = 40% is far above the correct 15%.

Common Pitfalls:Miscomputing the discriminant or forgetting that the inequality includes equality at the roots.

Final Answer:3/20