Solving (p^2-2p-8)/(p-1) ≥ 0: A Step-by-Step Guide

by Andrew McMorgan 51 views

Hey guys! Today, we're diving into a fun little inequality problem: (p^2 - 2p - 8) / (p - 1) ≥ 0. Don't worry, it's not as scary as it looks! We'll break it down step-by-step and get you feeling confident about solving these types of problems. So, grab your pencils, and let's get started!

1. Factoring the Quadratic Expression

The first thing we need to do is factor the quadratic expression in the numerator, which is p^2 - 2p - 8. Factoring quadratics is a crucial skill in algebra, and it's all about finding two numbers that multiply to give you the constant term (in this case, -8) and add up to the coefficient of the linear term (in this case, -2).

Think of it like this: We need two numbers, let's call them 'a' and 'b', such that:

  • a * b = -8
  • a + b = -2

After a bit of thought, you'll realize that the numbers -4 and 2 fit the bill perfectly. -4 multiplied by 2 is indeed -8, and -4 plus 2 is -2. Therefore, we can factor the quadratic expression as follows:

p^2 - 2p - 8 = (p - 4)(p + 2)

So, our inequality now looks like this:

((p - 4)(p + 2)) / (p - 1) ≥ 0

Factoring is a foundational step, as it allows us to identify the critical points of the inequality, which we'll use to determine the intervals where the expression is positive, negative, or zero.

2. Identifying Critical Points

Critical points are the values of p that make either the numerator or the denominator of the rational expression equal to zero. These points are critical because they are the locations where the expression can change its sign (from positive to negative or vice versa). Let's find those critical points:

  • Numerator: (p - 4)(p + 2) = 0. This gives us two critical points: p = 4 and p = -2.
  • Denominator: p - 1 = 0. This gives us one critical point: p = 1.

So, our critical points are p = -2, p = 1, and p = 4. These are the values of p where the expression might change its sign. Always remember that the values that make the denominator zero are crucial because they define where the function is undefined. We can't include them in our solution if the inequality is strict (i.e., > or <), but in this case, we need to consider them since it is greater than or equal to 0.

3. Creating a Sign Chart

A sign chart is a visual tool that helps us determine the sign of the expression in each interval defined by our critical points. It's essentially a number line divided into sections by the critical points. Here’s how we create one:

  1. Draw a number line and mark the critical points (-2, 1, and 4) on it. These points divide the number line into four intervals: (-∞, -2), (-2, 1), (1, 4), and (4, ∞).
  2. Choose a test value within each interval and plug it into the factored inequality ((p - 4)(p + 2)) / (p - 1). The sign of the result will tell you the sign of the expression in that entire interval.

Let's do it:

  • Interval (-∞, -2): Let's pick p = -3. Then, ((-3 - 4)(-3 + 2)) / (-3 - 1) = ((-7)(-1)) / (-4) = 7 / -4 = Negative.
  • Interval (-2, 1): Let's pick p = 0. Then, ((0 - 4)(0 + 2)) / (0 - 1) = ((-4)(2)) / (-1) = -8 / -1 = Positive.
  • Interval (1, 4): Let's pick p = 2. Then, ((2 - 4)(2 + 2)) / (2 - 1) = ((-2)(4)) / (1) = -8 / 1 = Negative.
  • Interval (4, ∞): Let's pick p = 5. Then, ((5 - 4)(5 + 2)) / (5 - 1) = ((1)(7)) / (4) = 7 / 4 = Positive.

Now we can create our sign chart:

Interval:   (-∞, -2)   (-2, 1)    (1, 4)    (4, ∞)
Test Value:    -3        0         2         5
Sign:         -         +         -         +

4. Determining the Solution Set

Okay, so we want to find where (p^2 - 2p - 8) / (p - 1) ≥ 0. This means we're looking for the intervals where the expression is positive or equal to zero. Looking at our sign chart, we can see that the expression is positive in the intervals (-2, 1) and (4, ∞).

Now, let's consider the critical points. The expression is equal to zero when the numerator is zero, which occurs at p = -2 and p = 4. Since our inequality includes "or equal to," we need to include these points in our solution. However, we must exclude the critical point p = 1 because it makes the denominator zero, and division by zero is undefined.

Therefore, the solution set in interval notation is:

[-2, 1) ∪ [4, ∞)

Key Considerations:

  • The square brackets [ ] indicate that the endpoint is included in the solution set.
  • The parenthesis ( ) indicate that the endpoint is excluded from the solution set.
  • The symbol ∪ represents the union of the two intervals.

5. Verification (Optional but Recommended)

To be absolutely sure we've got the correct solution, it's always a good idea to verify our answer. Choose a value from each interval in our solution and plug it back into the original inequality. If the inequality holds true, then our solution is likely correct.

  • Let's test p = 0 (from the interval [-2, 1)): ((0)^2 - 2(0) - 8) / (0 - 1) = (-8) / (-1) = 8 ≥ 0 (True)
  • Let's test p = 5 (from the interval [4, ∞)): ((5)^2 - 2(5) - 8) / (5 - 1) = (25 - 10 - 8) / (4) = 7 / 4 ≥ 0 (True)

Since the inequality holds true for our test values, we can be confident that our solution set is correct!

Conclusion

So, there you have it! The solution to the inequality (p^2 - 2p - 8) / (p - 1) ≥ 0 is [-2, 1) ∪ [4, ∞). Remember the key steps: factor the quadratic expression, identify the critical points, create a sign chart, and then determine the solution set based on the sign chart and the inequality. Understanding these steps will empower you to solve a wide range of inequalities!

Keep practicing, and you'll become a pro in no time. You got this! And remember guys, maths can be fun if you take it step by step. Keep rocking!