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: Understanding the Concept
The problem asks for the probability of an event. This is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. Here, the outcomes are the integers from 1 to 50, and a favorable outcome is an integer that satisfies the given inequality.
Step 2: Key Formula or Approach
1. Identify the total number of possible outcomes, which is the size of the set. 2. Solve the inequality $x + \frac{10}{x} \le 11$ for the integer values of $x$ in the given set. 3. Count the number of favorable outcomes. 4. Calculate the probability: $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.
Step 3: Detailed Explanation
The set of numbers is $S = \{1, 2, 3, \dots, 50\}$. 1. Total number of outcomes: The total number of possible outcomes is the number of elements in the set $S$, which is $N = 50$. 2. Solve the inequality: We need to find the number of integers $x \in S$ that satisfy $x + \frac{10}{x} \le 11$. Since $x$ is a positive integer, we can multiply the entire inequality by $x$ without changing the direction of the inequality sign. \[ x^2 + 10 \le 11x \] \[ x^2 - 11x + 10 \le 0 \] Factor the quadratic expression: \[ (x-1)(x-10) \le 0 \] This is a downward-opening parabola. The expression is less than or equal to zero between its roots (inclusive). The roots are $x=1$ and $x=10$. So, the solution to the inequality is $1 \le x \le 10$. 3. Count the number of favorable outcomes: We need to find the integers in the set $S$ that satisfy $1 \le x \le 10$. These are the integers $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. The number of favorable outcomes is 10. 4. Calculate the probability: \[ P(\text{favorable event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{10}{50} = \frac{1}{5} \] Step 4: Final Answer
The probability that the drawn number $x$ satisfies the inequality is $\frac{10}{50} = \frac{1}{5}$.