- The quadratic expression factors into \((x - 3)(x - 5) < 0\).
- Testing points within the intervals \((-\infty, 3)\), \((3, 5)\), and \((5, \infty)\) determines the sign of the product.
- Only the interval \((3, 5)\) produces a negative value.
- The correct option is \(\boxed{c}\), which corresponds to the solution set \(\boxed{(3, 5)}\).
Explanation
- Analyzing the Quadratic Inequality
The problem requires solving the quadratic inequality \(x^2 - 8x + 15 < 0\). We are given that this can be factored as \((x - 3)(x - 5) < 0\). To solve this, we must determine the range of values for \(x\) that make this product negative. A product of two factors is negative only if one factor is positive and the other is negative. The critical points where the expression equals zero are \(x = 3\) and \(x = 5\). These points divide the real number line into three distinct intervals: \((-\infty, 3)\), \((3, 5)\), and \((5, \infty)\).
- Testing Intervals
To find where the inequality \((x - 3)(x - 5) < 0\) is satisfied, we test a representative value from each of the three intervals:
- For the interval \((-\infty, 3)\), let's choose \(x = 0\). The expression becomes \((0 - 3)(0 - 5) = (-3)(-5) = 15\). Since \(15 > 0\), this interval does not satisfy the inequality.
- For the interval \((3, 5)\), let's choose \(x = 4\). The expression becomes \((4 - 3)(4 - 5) = (1)(-1) = -1\). Since \(-1 < 0\), this interval satisfies the inequality.
- For the interval \((5, \infty)\), let's choose \(x = 6\). The expression becomes \((6 - 3)(6 - 5) = (3)(1) = 3\). Since \(3 > 0\), this interval does not satisfy the inequality.
- Conclusion
Based on our testing, the only interval that satisfies the condition \((x - 3)(x - 5) < 0\) is \((3, 5)\). Comparing this to the given choices:
A\((-\infty, 3)\)
B\((5, \infty)\)
C\((3, 5)\)
The correct option is (c). Therefore, the solution to the inequality \(x^2 - 8x + 15 < 0\) is \(\boxed{(3, 5)}\).
Examples
Understanding how to solve quadratic inequalities is useful in many real-world applications, such as optimizing physical or economic systems. For instance, if the cost function of producing a certain good is given by a quadratic equation, solving an inequality can help find the range of production levels where the company stays within a certain budget or achieves profitability. This mathematical approach allows for informed decision-making in manufacturing, logistics, and finance.