Solving The Inequality: (x-5)(2x+3)/(x-1)^2 ≤ 0

Alright guys, let's dive into solving this inequality: $\frac{(x-5)(2 x+3)}{(x-1)^2} \leq 0$. Inequalities like these pop up all the time in mathematics, especially in calculus and analysis. Understanding how to solve them is super important for grasping more complex concepts later on. We're going to break this down step-by-step so you can tackle similar problems with confidence.

Understanding the Problem

First off, let's understand what the inequality is asking. We're looking for all the values of x that make the expression $\frac{(x-5)(2 x+3)}{(x-1)^2}$ less than or equal to zero. This means we want to find where this fraction is either negative or zero. The key here is to consider the sign of each factor in the numerator and the denominator.

The numerator has two factors: (x - 5) and (2x + 3). The denominator has one factor: (x - 1)^2. We need to figure out when each of these factors is positive, negative, or zero. The sign of the entire fraction will depend on the combination of these signs. Remember, a fraction is negative if the numerator and denominator have opposite signs, and it's positive if they have the same sign. Also, the fraction is zero when the numerator is zero (and the denominator isn't).

One thing to keep a close eye on is the denominator. Since it's squared, (x - 1)^2 will always be positive or zero. However, we have to exclude the values of x that make the denominator zero because division by zero is a big no-no in mathematics. So, x = 1 will be a critical point to exclude from our solution set.

Finding Critical Points

Critical points are the values of x that make either the numerator or the denominator equal to zero. These points are crucial because they divide the number line into intervals where the expression's sign remains constant. Let's find them:

1. x - 5 = 0 gives us x = 5
2. 2x + 3 = 0 gives us x = -3/2 = -1.5
3. (x - 1)^2 = 0 gives us x = 1

So, our critical points are x = -1.5, x = 1, and x = 5. These points will help us determine the intervals where the inequality holds true.

Creating a Sign Chart

Now, let's create a sign chart to analyze the intervals determined by our critical points. A sign chart helps us visualize where each factor is positive or negative. We'll consider the intervals: (-∞, -1.5), (-1.5, 1), (1, 5), and (5, ∞).

Here's how we fill out the chart:

• x - 5: This factor is negative when x < 5 and positive when x > 5.
• 2x + 3: This factor is negative when x < -1.5 and positive when x > -1.5.
• (x - 1)^2: This factor is always positive (except at x = 1, where it's zero, but we exclude that point).

Now, we multiply the signs in each interval to find the sign of the entire expression. For example, in the interval x < -1.5, we have (-)(-)/(+) = (+). Similarly, in the interval -1.5 < x < 1, we have (-)(+)/(+) = (-). And so on.

Determining the Solution

We want the intervals where the expression $\frac{(x-5)(2 x+3)}{(x-1)^2}$ is less than or equal to zero. From our sign chart, we see that this happens in the intervals (-1.5, 1) and (1, 5).

However, we also need to consider where the expression is equal to zero. This occurs when the numerator is zero, which is at x = -1.5 and x = 5. So we include these points in our solution.

Also, remember that x = 1 is not included because the denominator becomes zero at that point. Thus, the solution is:

$$[-1.5, 1) \cup (1, 5]$$

In interval notation, this means all values of x from -1.5 to 5, including -1.5 and 5, but excluding 1. This is the range of x values that satisfy the given inequality.

Final Answer

The solution to the inequality $\frac{(x-5)(2 x+3)}{(x-1)^2} \leq 0$ is:

$$x \in \left[-\frac{3}{2}, 1\right) \cup (1, 5]$$

So there you have it! We've successfully solved the inequality by finding critical points, creating a sign chart, and determining the intervals where the expression is negative or zero. Keep practicing these steps, and you'll become a pro at solving inequalities in no time!

Additional Tips and Tricks

When solving inequalities, keep these additional tips and tricks in mind to make the process smoother:

• Simplify First: Before you start, simplify the inequality as much as possible. Combine like terms, factor expressions, and clear any fractions if you can. This will make the inequality easier to work with and reduce the chances of making mistakes.
• Beware of Multiplying by Negatives: When multiplying or dividing both sides of an inequality by a negative number, remember to flip the direction of the inequality sign. For example, if you have -x < 3, multiplying both sides by -1 gives x > -3.
• Check Your Work: After you've found the solution, it's always a good idea to check your work by plugging in values from each interval into the original inequality. This will help you verify that your solution is correct and catch any errors you may have made.
• Graphing: Sketching a quick graph of the expression can often provide a visual confirmation of your solution. It’s not always necessary, but it can be a great way to double-check your work and ensure that you’re on the right track. Use a graphing calculator or an online tool to plot the function and see where it falls below or equals zero.
• Consider Edge Cases: Always be mindful of edge cases, such as when the expression equals zero or when the denominator is zero. These cases can often be the trickiest parts of solving inequalities, so make sure to pay close attention to them. Remember to include the points where the expression equals zero if the inequality includes an "equal to" sign, but exclude any points where the denominator is zero.

Common Mistakes to Avoid

Solving inequalities can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

• Forgetting to Flip the Inequality Sign: As mentioned earlier, always remember to flip the direction of the inequality sign when multiplying or dividing both sides by a negative number.
• Ignoring the Denominator: Don't forget to consider the denominator of any fractions in the inequality. The denominator cannot be zero, so you need to exclude any values of x that make the denominator zero from your solution.
• Incorrectly Combining Intervals: When combining intervals, make sure to pay attention to whether the endpoints should be included or excluded. Use square brackets to include an endpoint and parentheses to exclude it.
• Not Checking Your Work: Always check your work by plugging in values from each interval into the original inequality. This will help you catch any errors and ensure that your solution is correct.
• Assuming Linearity: Avoid assuming that inequalities behave linearly, especially when dealing with non-linear functions. Always rely on a sign chart or test values to determine the correct intervals.

Real-World Applications

Solving inequalities isn't just a theoretical exercise; it has many practical applications in various fields, including:

• Engineering: Engineers use inequalities to design structures that can withstand certain loads or stresses. They also use them to optimize the performance of systems and ensure that they meet certain specifications.
• Economics: Economists use inequalities to model and analyze economic phenomena, such as supply and demand, market equilibrium, and consumer behavior. They also use them to make predictions about future economic conditions.
• Computer Science: Computer scientists use inequalities to analyze the performance of algorithms and data structures. They also use them to design efficient systems and ensure that they meet certain performance requirements.
• Finance: Financial analysts use inequalities to assess risk and make investment decisions. They also use them to model and analyze financial markets and make predictions about future market conditions.

Inequalities are versatile tools that find application in numerous real-world scenarios, making their understanding crucial for students and professionals alike. By mastering the art of solving inequalities, you equip yourself with a valuable skill applicable across diverse domains.

Practice Problems

To reinforce your understanding of solving inequalities, here are a few practice problems for you to tackle:

1. Solve the inequality: $\frac{(x+2)(x-3)}{x-1} > 0$
2. Solve the inequality: $x^2 - 5x + 6 \leq 0$
3. Solve the inequality: $\frac{2x+1}{x-4} \geq 1$
4. Solve the inequality: $|x-3| < 2$

Work through these problems, and remember to follow the steps we discussed earlier: find the critical points, create a sign chart, and determine the intervals where the inequality holds true. Good luck, and happy solving!

By mastering these concepts and practicing regularly, you’ll be well-equipped to tackle even the most challenging inequality problems. Keep pushing forward, and don't hesitate to seek help when needed. Happy studying, and best of luck with your mathematical journey!