Solving The Inequality (x-3)(x+5) ≤ 0: A Step-by-Step Guide

Hey guys! Today, we're going to dive into solving the inequality $(x-3)(x+5) \leq 0$. This is a common type of problem in algebra, and understanding how to solve it will definitely help you out. We'll break it down step by step so it's super easy to follow. Let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have an inequality: $(x-3)(x+5) \leq 0$. Our goal is to find all the values of $x$ that make this inequality true. In other words, we want to find the range of $x$ values for which the expression $(x-3)(x+5)$ is either negative or zero.

Why is this important? Inequalities like this pop up in various areas of math and science, from calculus to physics. Being able to solve them quickly and accurately is a valuable skill. Plus, it's a great exercise for your brain!

Step 1: Find the Critical Points

The first thing we need to do is find the critical points. These are the values of $x$ that make the expression $(x-3)(x+5)$ equal to zero. To find them, we set each factor equal to zero:

1. $x - 3 = 0$
2. $x + 5 = 0

Solving these equations is pretty straightforward:

1. $x = 3$
2. $x = -5

So, our critical points are $x = 3$ and $x = -5$. These points are crucial because they divide the number line into intervals where the expression $(x-3)(x+5)$ will have a consistent sign (either positive or negative).

Critical points are incredibly important because they are the points where the function can change signs. In the context of inequalities, this means the critical points are the potential boundaries of our solution intervals. Imagine a rollercoaster; these points are like the peaks and valleys where the ride changes direction.

Step 2: Create a Sign Chart

Now that we have our critical points, we'll create a sign chart. A sign chart helps us visualize where the expression $(x-3)(x+5)$ is positive, negative, or zero. Here's how we set it up:

1. Draw a number line.
2. Mark the critical points ($x = -5$ and $x = 3$) on the number line.
3. These points divide the number line into three intervals: $(-\infty, -5)$, $(-5, 3)$, and $(3, \infty)$.
4. Choose a test value from each interval and plug it into the expression $(x-3)(x+5)$ to determine the sign of the expression in that interval.

Let's pick our test values:

• For $(-\infty, -5)$, let's choose $x = -6$.
• For $(-5, 3)$, let's choose $x = 0$.
• For $(3, \infty)$, let's choose $x = 4$.

Now, let's evaluate the expression $(x-3)(x+5)$ at these test values:

• For $x = -6$: $(-6-3)(-6+5) = (-9)(-1) = 9$ (positive)
• For $x = 0$: $(0-3)(0+5) = (-3)(5) = -15$ (negative)
• For $x = 4$: $(4-3)(4+5) = (1)(9) = 9$ (positive)

We can now complete our sign chart:

Step 3: Determine the Solution Set

We're looking for where $(x-3)(x+5) \leq 0$, which means we want the intervals where the expression is either negative or zero. From our sign chart, we see that the expression is negative in the interval $(-5, 3)$. We also need to include the critical points $x = -5$ and $x = 3$ because the inequality includes "equal to zero".

Therefore, the solution set is $-5 \leq x \leq 3$.

In set notation, this is written as ${x \mid -5 \leq x \leq 3}$.

Understanding the solution set is crucial. It represents all the values of x that satisfy the original inequality. In real-world terms, this could represent a range of acceptable values in a system or experiment. Think of it as the sweet spot where everything works as expected.

Why Other Options Are Incorrect

Let's quickly look at why the other options are incorrect:

• A. ${x \mid 3 \leq x \leq 5}$: This is wrong because it only considers values greater than or equal to 3, and it misses the interval where the expression is negative.
• B. ${x \mid -5 \leq x \leq -3}$: This is also incorrect. While it includes part of the correct interval, it doesn't extend to the correct upper bound (3).
• D. ${x \mid -3 \leq x \leq 5}$: This is wrong because it includes values outside the correct interval and misses the lower bound.

Final Answer

The correct answer is C. ${x \mid -5 \leq x \leq 3}$. This represents all the values of $x$ for which the inequality $(x-3)(x+5) \leq 0$ is true.

Pro Tip: Always double-check your answer by plugging in values from your solution set into the original inequality. This helps ensure you haven't made any mistakes.

Wrapping Up

So, there you have it! Solving inequalities like $(x-3)(x+5) \leq 0$ involves finding critical points, creating a sign chart, and determining the solution set. With a little practice, you'll become a pro at these types of problems. Keep practicing, and you'll ace those math tests in no time! You got this!

Remember, the key to mastering these problems is practice, practice, practice! The more you work through them, the more comfortable you'll become with the process. And don't be afraid to ask for help if you get stuck. There are plenty of resources available, including textbooks, online tutorials, and, of course, your friendly neighborhood math teacher.

Final Thoughts: Math might seem intimidating, but with the right approach and a bit of perseverance, anyone can conquer it. So keep your chin up, stay positive, and never stop learning.

Happy solving, and I'll catch you in the next explanation!