Questions: Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. (x + 1)(x - 4)(x + 5) ≤ 0 Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution set is [-5,4] (Simplify your answer. Use interval notation. Use integers or fractions for any numbers in the expression.) The solution set is the empty set Which number line shows the graph of the solution set? A. ---------------------------------------------------- -6 -4 -2 0 2 4 6 B. ---------------------------------------------------- -6 -4 -2 0 2 4 6 C. ---------------------------------------------------- -6 -4 -2 0 2 4 6 D. ---------------------------------------------------- -6 -4 -2 0 2 4 6

Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. (x + 1)(x - 4)(x + 5) ≤ 0 Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution set is [-5,4] (Simplify your answer. Use interval notation. Use integers or fractions for any numbers in the expression.) The solution set is the empty set Which number line shows the graph of the solution set? A. ---------------------------------------------------- -6 -4 -2 0 2 4 6 B. ---------------------------------------------------- -6 -4 -2 0 2 4 6 C. ---------------------------------------------------- -6 -4 -2 0 2 4 6 D. ---------------------------------------------------- -6 -4 -2 0 2 4 6
Transcript text: Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. (x + 1)(x - 4)(x + 5) ≤ 0 Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The solution set is [-5,4] (Simplify your answer. Use interval notation. Use integers or fractions for any numbers in the expression.) The solution set is the empty set Which number line shows the graph of the solution set? A. --|--------|--------|--------|--------|--------|--------|-- -6 -4 -2 0 2 4 6 B. --|--------|--------|--------|--------|--------|--------|-- -6 -4 -2 0 2 4 6 C. --|--------|--------|--------|--------|--------|--------|-- -6 -4 -2 0 2 4 6 D. --|--------|--------|--------|--------|--------|--------|-- -6 -4 -2 0 2 4 6
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Solution

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Solution Steps

Step 1: Find the critical points

To solve the inequality \((x + 1)(x - 4)(x + 5) \leq 0\), we first find the critical points by setting each factor equal to zero: \[ x + 1 = 0 \implies x = -1 \] \[ x - 4 = 0 \implies x = 4 \] \[ x + 5 = 0 \implies x = -5 \]

Step 2: Determine the intervals

The critical points divide the real number line into four intervals: \[ (-\infty, -5), (-5, -1), (-1, 4), (4, \infty) \]

Step 3: Test the intervals

We test a point in each interval to determine where the product is less than or equal to zero.

  1. For \(x \in (-\infty, -5)\), choose \(x = -6\): \[ (-6 + 1)(-6 - 4)(-6 + 5) = (-5)(-10)(-1) = -50 \quad (\text{negative}) \]

  2. For \(x \in (-5, -1)\), choose \(x = -3\): \[ (-3 + 1)(-3 - 4)(-3 + 5) = (-2)(-7)(2) = 28 \quad (\text{positive}) \]

  3. For \(x \in (-1, 4)\), choose \(x = 0\): \[ (0 + 1)(0 - 4)(0 + 5) = (1)(-4)(5) = -20 \quad (\text{negative}) \]

  4. For \(x \in (4, \infty)\), choose \(x = 5\): \[ (5 + 1)(5 - 4)(5 + 5) = (6)(1)(10) = 60 \quad (\text{positive}) \]

Final Answer

The solution set is \([-5, -1] \cup [-1, 4]\).

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