To solve the inequality [tex]\((x - 3)(x + 5) \leq 0\)[/tex], we will carefully go through each step in detail.
1. Find the roots of the equation:
The inequality [tex]\((x - 3)(x + 5) \leq 0\)[/tex] can be split into finding the roots of the equation [tex]\((x - 3)(x + 5) = 0\)[/tex]. Solving this:
[tex]\[
x - 3 = 0 \quad \text{or} \quad x + 5 = 0
\][/tex]
The solutions are:
[tex]\[
x = 3 \quad \text{and} \quad x = -5
\][/tex]
2. Determine the intervals:
These roots split the number line into three intervals. We need to determine the sign of [tex]\((x - 3)(x + 5)\)[/tex] in each interval:
- Interval 1: [tex]\(x < -5\)[/tex]
- Interval 2: [tex]\(-5 \leq x \leq 3\)[/tex]
- Interval 3: [tex]\(x > 3\)[/tex]
3. Sign Analysis in the intervals:
- For [tex]\(x < -5\)[/tex]:
Choose a test point, e.g., [tex]\(x = -6\)[/tex]:
[tex]\((x - 3)(x + 5) = (-6 - 3)(-6 + 5) = (-9)(-1) = 9\)[/tex], which is positive.
- For [tex]\(-5 \leq x \leq 3\)[/tex]:
Choose a test point, e.g., [tex]\(x = 0\)[/tex]:
[tex]\((x - 3)(x + 5) = (0 - 3)(0 + 5) = (-3)(5) = -15\)[/tex], which is negative.
- For [tex]\(x > 3\)[/tex]:
Choose a test point, e.g., [tex]\(x = 4\)[/tex]:
[tex]\((x - 3)(x + 5) = (4 - 3)(4 + 5) = (1)(9) = 9\)[/tex], which is positive.
4. Include the roots:
Since the inequality is [tex]\(\leq 0\)[/tex], and we have [tex]\((x - 3)(x + 5) = 0\)[/tex] at [tex]\(x = 3\)[/tex] and [tex]\(x = -5\)[/tex], we need to include these points since the inequality allows zero.
5. Form the solution:
The values of [tex]\(x\)[/tex] that satisfy [tex]\((x - 3)(x + 5) \leq 0\)[/tex] are within the interval where the expression is negative, including the points where it equals zero.
Thus, putting this all together, the solution to the inequality [tex]\((x-3)(x+5) \leq 0\)[/tex] is:
[tex]\[
-5 \leq x \leq 3
\][/tex]
Therefore, the correct answer from the given choices is:
[tex]\[
\{x \mid -5 \leq x \leq 3\}
\][/tex]
