Let `A cup B = {x|(x-a)(x-b)gt0`, where `a lt b`}, what are A and B equal to?

Options

A. `A={x|xgta}andB={x|xgtb}`
B. `A={x{|xlta}andB={x|xgtb}`
C. `A={x|xlta}andB={x|xltb}`
D. `A={x|xgta}andB={x|xltb}`

Answer

To solve the problem, we need to analyze the given expression \( A \cup B = \{ x \mid (x - a)(x - b) > 0 \} \), where \( a < b \).

### Step 1: Understand the inequality
The inequality \( (x - a)(x - b) > 0 \) indicates that the product of the two factors is positive. This can happen in two scenarios:
1. Both factors are positive: \( x - a > 0 \) and \( x - b > 0 \)
2. Both factors are negative: \( x - a < 0 \) and \( x - b < 0 \)

### Step 2: Analyze the first scenario
For the first scenario where both factors are positive:
- \( x - a > 0 \) implies \( x > a \)
- \( x - b > 0 \) implies \( x > b \)

Since \( a < b \), if \( x > b \), then it is also true that \( x > a \). Therefore, in this case, the solution set is:
\[ x > b \]

### Step 3: Analyze the second scenario
For the second scenario where both factors are negative:
- \( x - a < 0 \) implies \( x < a \)
- \( x - b < 0 \) implies \( x < b \)

Since \( a < b \), if \( x < a \), then it is also true that \( x < b \). Therefore, in this case, the solution set is:
\[ x < a \]

### Step 4: Combine the results
Combining the results from both scenarios, we find that:
- The solution set for \( (x - a)(x - b) > 0 \) is:
\[ x < a \quad \text{or} \quad x > b \]

### Step 5: Define sets A and B
From the analysis, we can define sets \( A \) and \( B \) as follows:
- Set \( A \) corresponds to the values where \( x < a \):
\[ A = \{ x \mid x < a \} \]
- Set \( B \) corresponds to the values where \( x > b \):
\[ B = \{ x \mid x > b \} \]

### Final Answer
Thus, the sets \( A \) and \( B \) are:
- \( A = \{ x \mid x < a \} \)
- \( B = \{ x \mid x > b \} \)

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