Understanding the Set Union Defined by an Inequality

The problem asks us to identify the individual sets A and B given their union, \(A \cup B\), is defined by the inequality \((x - a)(x - b) > 0\), where it is specified that \(a < b\). To solve this, we first need to understand the set described by the given inequality.

Solving the Inequality \((x - a)(x - b) > 0\)

The inequality \((x - a)(x - b) > 0\) means that the product of the two factors, \((x - a)\) and \((x - b)\), must be positive. This can happen in two possible ways:

1. Both factors are positive.
2. Both factors are negative.

Case 1: Both factors are positive

• \(x - a > 0 \implies x > a\)
• \(x - b > 0 \implies x > b\)

Since we are given that \(a < b\), for both conditions \(x > a\) and \(x > b\) to be true simultaneously, x must be greater than the larger value, which is \(b\). So, this case gives us the condition \(x > b\).

Case 2: Both factors are negative

• \(x - a < 0 \implies x < a\)
• \(x - b < 0 \implies x < b\)

Since we are given that \(a < b\), for both conditions \(x < a\) and \(x < b\) to be true simultaneously, x must be less than the smaller value, which is \(a\). So, this case gives us the condition \(x < a\).

Combining the conditions from both cases, the solution to the inequality \((x - a)(x - b) > 0\) is \(x < a\) or \(x > b\).

Therefore, the given set \(A \cup B\) is equal to \(\{x \mid x < a \text{ or } x > b\}\).

Analyzing the Options for Sets A and B

We are given four possible pairs of sets A and B. We need to find the pair whose union matches the set \(\{x \mid x < a \text{ or } x > b\}\).

As shown in the table, only the second option, where \(A = \{x \mid x < a\}\) and \(B = \{x \mid x > b\}\), results in the union \(A \cup B = \{x \mid x < a \text{ or } x > b\}\).

Therefore, the correct definitions for sets A and B are \(A = \{x \mid x < a\}\) and \(B = \{x \mid x > b\}\).

Revision Table: Key Concepts in Set Theory and Inequalities

Additional Information on Solving Quadratic Inequalities

The inequality \((x - a)(x - b) > 0\) is a simple form of a quadratic inequality. Quadratic inequalities involving a quadratic expression \(px^2 + qx + r\) can often be solved by factoring the quadratic expression into the form \(p(x - \alpha)(x - \beta)\) or by finding the roots of the corresponding quadratic equation \(px^2 + qx + r = 0\).

The roots (\(\alpha\) and \(\beta\)) divide the number line into intervals. The sign of the quadratic expression \(p(x - \alpha)(x - \beta)\) is constant within each interval.

For an inequality like \((x - a)(x - b) > 0\) with roots \(a\) and \(b\) and \(a < b\), the number line is divided into three intervals: \((-\infty, a)\), \((a, b)\), and \((b, \infty)\).

• In \((-\infty, a)\), both \((x-a)\) and \((x-b)\) are negative, so \((x-a)(x-b)\) is positive.
• In \((a, b)\), \((x-a)\) is positive and \((x-b)\) is negative, so \((x-a)(x-b)\) is negative.
• In \((b, \infty)\), both \((x-a)\) and \((x-b)\) are positive, so \((x-a)(x-b)\) is positive.

Therefore, \((x - a)(x - b) > 0\) holds for \(x \in (-\infty, a) \cup (b, \infty)\), which is the set \(\{x \mid x < a \text{ or } x > b\}\).