Question

If a number x is drawn randomly from the set of numbers \{1, 2, 3, ..., 50\}, then the probability that number x that is drawn satisfies the inequation $x + \frac{10}{x} \le 11$ is

• A. $\frac{4}{5}$
• B. $\frac{9}{50}$
• C. $\frac{4}{25}$
• D. $\frac{1}{5}$

Hint:

When solving inequalities involving rational expressions like $f(x)/g(x)$, be careful when multiplying by a term containing the variable. You can only do this without considering cases if you are certain of the sign of the term. In this problem, since $x$ is from a set of positive integers, we know $x>0$, so multiplying by $x$ is safe.

Answer 1

The Correct Option is D

Step 1: Solve the Inequality: \[ x + \frac{10}{x} \le 11 \] Since \( x \in \{1, \dots, 50\} \), \( x>0 \). Multiply by \( x \): \[ x^2 + 10 \le 11x \] \[ x^2 - 11x + 10 \le 0 \] Factor the quadratic: \[ (x-1)(x-10) \le 0 \] The solution to this inequality is \( 1 \le x \le 10 \).
Step 2: Count Favorable Outcomes: The integers in the set \( \{1, 2, \dots, 50\} \) that satisfy \( 1 \le x \le 10 \) are \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \). Number of favorable outcomes = 10.
Step 3: Calculate Probability: Total outcomes = 50. \[ P(E) = \frac{10}{50} = \frac{1}{5} \]