To solve the problem, we need to find the probability \( P(A) \) that a randomly chosen number \( x \) from the set \( \{1, 2, 3, \ldots, 100\} \) satisfies the inequality:
\[
\frac{(x-10)(x-50)}{(x-30)} \geq 0
\]
### Step 1: Identify Critical Points
The critical points of the inequality are found by setting the numerator and denominator to zero:
1. \( x - 10 = 0 \) → \( x = 10 \)
2. \( x - 50 = 0 \) → \( x = 50 \)
3. \( x - 30 = 0 \) → \( x = 30 \)
Thus, the critical points are \( x = 10, 30, 50 \).
### Step 2: Test Intervals
Next, we will test the intervals defined by these critical points to determine where the inequality holds true. The intervals are:
- \( (-\infty, 10) \)
- \( (10, 30) \)
- \( (30, 50) \)
- \( (50, \infty) \)
We will choose test points from each interval:
1. For \( x < 10 \) (e.g., \( x = 0 \)):
\[
\frac{(0-10)(0-50)}{(0-30)} = \frac{(−10)(−50)}{(−30)} = \frac{500}{−30} < 0
\]
2. For \( 10 < x < 30 \) (e.g., \( x = 20 \)):
\[
\frac{(20-10)(20-50)}{(20-30)} = \frac{(10)(−30)}{(−10)} = \frac{−300}{−10} > 0
\]
3. For \( 30 < x < 50 \) (e.g., \( x = 40 \)):
\[
\frac{(40-10)(40-50)}{(40-30)} = \frac{(30)(−10)}{(10)} = \frac{−300}{10} < 0
\]
4. For \( x > 50 \) (e.g., \( x = 60 \)):
\[
\frac{(60-10)(60-50)}{(60-30)} = \frac{(50)(10)}{(30)} > 0
\]
### Step 3: Determine Valid Intervals
From the tests, we conclude:
- The inequality is satisfied in the intervals:
- \( [10, 30) \) (includes 10, excludes 30)
- \( (50, \infty) \)
However, since \( x \) can only be chosen from \( \{1, 2, \ldots, 100\} \), we restrict the second interval to \( (50, 100] \).
### Step 4: Count Valid Values of \( x \)
Now, we count the valid values of \( x \):
1. From \( 10 \) to \( 29 \) (inclusive):
- Total values = \( 29 - 10 + 1 = 20 \)
2. From \( 51 \) to \( 100 \) (inclusive):
- Total values = \( 100 - 51 + 1 = 50 \)
Adding these gives us:
\[
20 + 50 = 70
\]
### Step 5: Calculate Total Possible Outcomes
The total number of possible outcomes (the total values of \( x \)) is \( 100 \).
### Step 6: Calculate Probability
The probability \( P(A) \) is given by the ratio of favorable outcomes to total outcomes:
\[
P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{70}{100} = 0.7
\]
### Final Answer
Thus, the probability \( P(A) \) is:
\[
\boxed{0.7}
\]
