Question

Solve the following inequalities using a sign diagram: 1−p /p +4≤p

Answer 1

To solve the inequality 1 - p / p + 4 ≤ p, we can start by finding the values of p that make the left side equal to or less than the right side. By considering three cases and creating a sign diagram, we find that the solutions are p ≤ -1.382 or 2.382 ≤ p.

Step-by-step explanation:

To solve the inequality 1 - p / p + 4 ≤ p, we can start by finding the values of p that make the left side equal to or less than the right side. We'll consider three cases: p + 4 > 0, p + 4 = 0, and p + 4 < 0.

1. For p + 4 > 0, the inequality becomes 1 - p / p + 4 ≤ p, which simplifies to 1 - 1 / p + 4 ≤ p. Multiplying both sides by p + 4, we get p + 4 - 1 ≤ p(p + 4), which can be simplified to p + 3 ≤ p² + 4p. Rearranging the terms, we have p² + 3p + 3 ≤ 4p. Moving all terms to one side, we get p² - p - 3 ≤ 0. To find the solutions, we can create a sign diagram to determine the sign of the expression p² - p - 3 for different values of p. By factoring or using the quadratic formula, we find that the solutions are approximately p ≤ -1.382 or 2.382 ≤ p.
2. For p + 4 = 0, we have p = -4. This value is not a solution to the original inequality.
3. For p + 4 < 0, the inequality becomes 1 - p / p + 4 ≤ p, which simplifies to 1 + p / p + 4 ≤ p. Multiplying both sides by p + 4, we get p + 4 + p ≤ p(p + 4), which can be simplified to 2p + 4 ≤ p² + 4p. Rearranging the terms, we have p² + 2p + 4 ≤ 0. This quadratic equation has no real solutions.

Combining the solutions from the three cases, we have p ≤ -1.382 or 2.382 ≤ p.