Solving Inequalities: A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon an inequality like $(x-3)(x+5) \leq 0$ and felt a little lost? Don't worry, it's totally normal! Inequalities might seem tricky at first, but with a clear understanding of the concepts and a systematic approach, you can crack them with ease. In this article, we'll break down the process of solving this specific inequality, step by step, making sure you understand every detail along the way. We will also look into the correct answer for the multiple-choice question: A. ${x \mid 3 \leq x \leq 5}$, B. ${x \mid-3 \leq x \leq 5}$, C. ${x \mid-5 \leq x \leq 3}$, D. ${x \mid-5 \leq x \leq-3}$. Let's dive in and demystify this mathematical puzzle!

Understanding the Basics: What Does the Inequality Mean?

Alright, before we jump into the solution, let's make sure we're all on the same page. The inequality $(x-3)(x+5) \leq 0$ is asking us a simple question: For what values of 'x' is the product of (x-3) and (x+5) less than or equal to zero? Remember, a product of two factors is negative (less than zero) when one factor is positive and the other is negative. It is zero when one or both factors are zero. So, our goal is to find the range of 'x' values that satisfy this condition. The crux of solving inequalities like this lies in understanding the behavior of the expression $(x-3)(x+5)$ across different intervals of 'x'. We are, essentially, detectives, trying to find out where this expression changes its sign. The sign of the expression is crucial because it tells us whether the product is positive, negative, or zero. When the product is negative or zero, the inequality is satisfied. The zero points, also known as roots, are the points at which the expression equals zero. In the case of $(x-3)(x+5)$, the roots are x = 3 and x = -5. These roots are critical because they divide the number line into intervals where the expression's sign is consistent. In other words, the sign of $(x-3)(x+5)$ will not change within any given interval unless we cross a root. Therefore, our strategy is to identify these roots and test the intervals defined by these roots. This systematic approach ensures that we capture all possible solutions to the inequality. This method of testing intervals is a reliable and efficient technique for tackling polynomial inequalities, providing a solid foundation for solving more complex problems. The value of x that makes each of the factors equal to zero are called the zeros of the expression. For $(x-3)(x+5)$, these values are x = 3 and x = -5.

Now, how do we find those intervals? Well, let's move on to the next section where we figure that out and finally solve the inequality.

Finding the Critical Points: The Zeros of the Expression

Before we delve into the heart of the solution, let's talk about the critical points. These are the values of 'x' where the expression $(x-3)(x+5)$ equals zero. These points are super important because they divide the number line into different intervals. Within each interval, the sign of the expression (whether it's positive or negative) remains constant. To find these critical points, we simply set each factor in the expression to zero and solve for 'x'.

So, we have two factors: $(x-3)$ and $(x+5)$.

1. For $(x-3) = 0$, solving for 'x' gives us $x = 3$. This is one of our critical points.
2. For $(x+5) = 0$, solving for 'x' gives us $x = -5$. This is our second critical point.

So, we have two critical points: $x = 3$ and $x = -5$. These points are the zeros of the expression. The zeros are the values of x that make the expression equal to zero. These critical points divide the number line into three intervals: $(-\infty, -5)$, $(-5, 3)$, and $(3, \infty)$. The values -5 and 3 are the roots of the equation. These roots help us define the intervals we will be testing to find the solution to the inequality. Therefore, the critical points, or zeros, are the foundation upon which we build our solution. Knowing these critical points allows us to split the number line into manageable intervals, making the process of solving the inequality more systematic and straightforward. We'll test each interval to see where the inequality holds true.

Now, let's see how these critical points help us find the solution.

Testing the Intervals: Where is the Expression Negative or Zero?

We've got our critical points, $x = -5$ and $x = 3$. Now, we need to test the intervals they create to see where the inequality $(x-3)(x+5) \leq 0$ is true. We'll consider three intervals: $(-\infty, -5)$, $(-5, 3)$, and $(3, \infty)$. To do this, we pick a test value within each interval and plug it into the expression $(x-3)(x+5)$. If the result is less than or equal to zero, the interval is part of our solution. So, let's get testing!

1. Interval $(-\infty, -5)$: Let's pick $x = -6$ (since $-6$ is less than $-5$). Substitute $x = -6$ into $(x-3)(x+5)$: $(-6-3)(-6+5) = (-9)(-1) = 9$. The result is positive (9 > 0). This interval is not part of the solution.

2. Interval $(-5, 3)$: Let's pick $x = 0$ (since $0$ is between $-5$ and $3$). Substitute $x = 0$ into $(x-3)(x+5)$: $(0-3)(0+5) = (-3)(5) = -15$. The result is negative $(-15 \leq 0)$. This interval is part of the solution.

3. Interval $(3, \infty)$: Let's pick $x = 4$ (since $4$ is greater than $3$). Substitute $x = 4$ into $(x-3)(x+5)$: $(4-3)(4+5) = (1)(9) = 9$. The result is positive $(9 > 0)$. This interval is not part of the solution.

So, the only interval that satisfies the inequality is $(-5, 3)$. But wait, we also need to consider the points where the expression equals zero. Those are our critical points, $x = -5$ and $x = 3$. Since the inequality includes 'equal to', we must include these points in our solution. Therefore, the solution to $(x-3)(x+5) \leq 0$ is the interval $-5 \leq x \leq 3$. By systematically testing each interval, we ensure that we capture all values of 'x' that satisfy the given inequality. This approach is a fundamental technique in solving a wide range of inequalities, from simple quadratics to more complex polynomial expressions. The value zero is very important since the inequality uses the symbol less than or equal to. This means that the zeros of the expression are part of the solution.

Let's recap our findings and pinpoint the correct answer!

The Solution: Putting It All Together

Alright, we've done the hard work, now let's put the pieces together! We tested the intervals, and we found that the expression $(x-3)(x+5)$ is negative or zero in the interval $(-5, 3)$, and we include -5 and 3 because the inequality is less than or equal to. The expression is zero at $x = -5$ and $x = 3$. These are the zeros of the expression. Therefore, the solution to the inequality $(x-3)(x+5) \leq 0$ is all the x values between -5 and 3, including -5 and 3. Thus, the solution is $-5 \leq x \leq 3$. This means x can be any value from -5 to 3, including -5 and 3. Graphically, this is represented by a closed interval on the number line, from -5 to 3, with closed circles at -5 and 3, indicating that these points are included in the solution. When solving inequalities, remember that the sign of the expression is the key. That sign depends on the factors of the expression. When a factor changes its sign, the whole expression changes its sign. When we identify the points at which the sign changes, we have defined the intervals that may be part of the solution. With a firm grasp of how to identify the intervals and test them, you're equipped to handle a variety of inequalities.

Now, let's match our solution to the multiple-choice options.

Identifying the Correct Answer

We've concluded that the solution to the inequality $(x-3)(x+5) \leq 0$ is $-5 \leq x \leq 3$. This means that 'x' can be any number between -5 and 3, including -5 and 3. Considering the options presented:

A. ${x \mid 3 \leq x \leq 5}$ - This option states that x is between 3 and 5, which is incorrect. B. ${x \mid-3 \leq x \leq 5}$ - This option states that x is between -3 and 5, which is incorrect. C. ${x \mid-5 \leq x \leq 3}$ - This option states that x is between -5 and 3. This matches our solution. This is the correct answer. D. ${x \mid-5 \leq x \leq-3}$ - This option states that x is between -5 and -3, which is incorrect.

Therefore, the correct answer is C. ${x \mid-5 \leq x \leq 3}$. Congratulations, you've successfully solved the inequality! You did great.

Conclusion

Solving inequalities might seem daunting at first, but as you've seen, it's a straightforward process when you break it down into smaller steps. We started by understanding the meaning of the inequality, then we found the critical points (zeros), tested the intervals, and finally identified the solution. Remember, practice makes perfect! The more inequalities you solve, the more confident you'll become. Keep up the great work, and happy solving!