Solving Quadratic Inequalities A Step-by-Step Guide

Hey guys! Today, we're diving into the world of quadratic inequalities. Specifically, we're going to tackle the inequality $x^2 + 3x - 17 ext{ ≥ } x - 2$. Don't worry, it might look intimidating at first, but we'll break it down into manageable steps. By the end of this guide, you'll be a pro at solving these types of problems and expressing your solutions in interval notation. So, let's get started!

Understanding Quadratic Inequalities

Before we jump into the solution, let's quickly recap what quadratic inequalities are all about. At its core, quadratic inequalities involve comparing a quadratic expression (something in the form of $ax^2 + bx + c$) to another value, often zero, using inequality symbols like ≥, >, ≤, or <. Think of it as an extension of solving quadratic equations, but instead of finding specific solutions, we are looking for ranges of values that satisfy the inequality.

The key concept here is that the solutions to a quadratic inequality are often intervals or unions of intervals on the number line. This is because the quadratic expression can be positive, negative, or zero depending on the value of x. Our goal is to identify the intervals where the inequality holds true. To effectively solve quadratic inequalities, you should grasp these fundamental concepts and be well-versed in algebraic manipulations.

The Importance of Interval Notation

Now, let's talk about interval notation. When we're dealing with inequalities, the solution isn't just a single number – it's usually a range of numbers. Interval notation is a neat way to express these ranges. For instance, if the solution includes all numbers greater than 2, we write it as $(2, ∞)$. If it includes 2 itself, we use a square bracket: $[2, ∞)$. Parentheses indicate that the endpoint is not included, while square brackets mean it is included. Mastering interval notation is essential for accurately representing the solution sets of inequalities and communicating mathematical solutions clearly and concisely. It's a fundamental skill in algebra and calculus, providing a standardized way to express ranges of values and intervals.

Step-by-Step Solution

Okay, let's get to the nitty-gritty and solve the inequality $x^2 + 3x - 17 ext{ ≥ } x - 2$. We'll go through each step carefully.

Step 1: Rearrange the Inequality

The first step is to rearrange the inequality so that one side is zero. This is crucial because it allows us to easily identify the critical points (the points where the quadratic expression equals zero). Subtract $(x - 2)$ from both sides of the inequality:

$x^2 + 3x - 17 - (x - 2) ext{ ≥ } 0$

Simplify the expression:

$x^2 + 3x - 17 - x + 2 ext{ ≥ } 0$

$x^2 + 2x - 15 ext{ ≥ } 0$

Now we have a standard quadratic inequality ready to be factored.

Step 2: Factor the Quadratic Expression

The next step involves factoring the quadratic expression. Factoring helps us find the values of x where the expression equals zero, which are the critical points that divide the number line into intervals. Let's factor $x^2 + 2x - 15$:

We are looking for two numbers that multiply to -15 and add up to 2. These numbers are 5 and -3.

So, we can factor the quadratic as follows:

$(x + 5)(x - 3) ext{ ≥ } 0$

Step 3: Find the Critical Points

Now, let's find the critical points. These are the values of x that make the factored expression equal to zero. They are crucial because they divide the number line into intervals where the quadratic expression will have a consistent sign (either positive or negative).

Set each factor to zero and solve for x:

$x + 5 = 0$ or $x - 3 = 0$

Solving these equations gives us the critical points:

$x = -5$ and $x = 3$

These points are the boundaries of our intervals.

Step 4: Create a Sign Chart

This is where things get visual! We're going to create a sign chart. A sign chart helps us determine the sign of the quadratic expression $(x + 5)(x - 3)$ in each interval created by the critical points. This will allow us to identify the intervals where the expression is greater than or equal to zero.

Draw a number line and mark the critical points, -5 and 3. This divides the number line into three intervals: $(-∞, -5)$, $(-5, 3)$, and $(3, ∞)$.

Choose a test value within each interval and plug it into each factor $(x + 5)$ and $(x - 3)$. Determine the sign of each factor in the interval. Then, multiply the signs to find the sign of the entire expression $(x + 5)(x - 3)$.

Step 5: Determine the Solution Set

Alright, we're almost there! Now we determine the solution set. We're looking for the intervals where $(x + 5)(x - 3) ext{ ≥ } 0$. This means we want the intervals where the expression is either positive or zero.

From our sign chart, we see that the expression is positive in the intervals $(-∞, -5)$ and $(3, ∞)$. It is also equal to zero at the critical points $x = -5$ and $x = 3$. Since our inequality includes “equal to,” we include these critical points in our solution.

Therefore, the solution set in interval notation is:

$(-∞, -5] ext{ ∪ } [3, ∞)$

We use square brackets to indicate that -5 and 3 are included in the solution.

Expressing the Solution in Interval Notation

As we've seen, expressing the solution in interval notation is the final step. Interval notation is a standardized way to write down intervals on the number line. Let’s recap the basics:

• Parentheses () are used to indicate that an endpoint is not included in the interval.
• Square brackets [] are used to indicate that an endpoint is included in the interval.
• The symbol ∞ represents infinity, and -∞ represents negative infinity. Infinity is always enclosed in parentheses because it is not a specific number.
• The union symbol ∪ is used to combine multiple intervals.

In our case, the solution set is all real numbers less than or equal to -5, as well as all real numbers greater than or equal to 3. This is why we write the solution as $(-∞, -5] ext{ ∪ } [3, ∞)$. This notation clearly and concisely represents the solution to the inequality.

Common Mistakes to Avoid

Let's chat about some common mistakes to avoid when solving quadratic inequalities. Knowing these pitfalls can save you a lot of headaches!

1. Forgetting to Rearrange the Inequality: One of the most common mistakes is trying to factor or analyze the inequality before setting it to zero. Always rearrange the inequality so that one side is zero first.
2. Incorrectly Factoring the Quadratic: Make sure you factor the quadratic expression correctly. Double-check your factors by expanding them to ensure they match the original expression.
3. Ignoring the Critical Points: The critical points are crucial! They define the intervals you need to test. Don't forget to find them and include them in your sign chart.
4. Using the Wrong Inequality Symbols: Pay close attention to the inequality symbol in the original problem. Use the correct symbols when expressing your solution in interval notation. A small oversight here can lead to an incorrect answer.
5. Not Using a Sign Chart: Trying to solve the inequality without a sign chart can be confusing and prone to errors. The sign chart provides a clear visual representation of the intervals and the signs of the factors.
6. Incorrect Interval Notation: Misusing parentheses and square brackets in interval notation is another common mistake. Remember, use square brackets if the endpoint is included, and parentheses if it’s not.

By keeping these common mistakes in mind, you'll be well-equipped to tackle quadratic inequalities accurately and confidently.

Practice Problems

To really nail down your understanding, let's look at a few practice problems. These will give you a chance to apply what you've learned and build your confidence. Remember, practice makes perfect!

1. Solve $x^2 - 4x + 3 < 0$
2. Solve $2x^2 + 5x - 12 ext{ ≤ } 0$
3. Solve $x^2 - 9 ext{ ≥ } 0$

Work through these problems step-by-step, following the method we've outlined. Check your solutions using a sign chart to ensure accuracy. If you get stuck, revisit the steps and examples we've discussed. The more you practice, the more comfortable you'll become with solving quadratic inequalities.

Conclusion

And there you have it! We've walked through the process of solving the quadratic inequality $x^2 + 3x - 17 ext{ ≥ } x - 2$ step-by-step. Remember, the key is to rearrange the inequality, factor the quadratic expression, find the critical points, use a sign chart, and express the solution in interval notation. Keep practicing, and you'll master this skill in no time. You've got this!