Solving Inequalities A Step-by-Step Guide To $\frac{6 X}{x^2-36} \geq 0$

Hey guys! Today, we're diving deep into the world of inequalities, specifically how to solve the inequality $\frac{6x}{x^2 - 36} \geq 0$. This type of problem often pops up in algebra and calculus, so mastering it is super important. We'll break it down step by step, making sure you understand the why behind each move, not just the how. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we fully understand the problem. Our main goal in solving inequalities is to find all the values of x that make the inequality true. In this case, we want to find the values of x for which the fraction $\frac{6x}{x^2 - 36}$ is greater than or equal to zero. This means the fraction can be positive or zero.

Now, let's take a closer look at the inequality itself. We have a rational expression, which is a fraction where the numerator and denominator are polynomials. Rational expressions can be tricky because the denominator cannot be zero (division by zero is a big no-no!). So, we need to keep that in mind as we solve.

The expression $x^2 - 36$ in the denominator is a difference of squares, which can be factored. Factoring is often a crucial step in solving inequalities, as it helps us identify the critical points where the expression might change its sign. The numerator, $6x$, is a simple linear term, which is also important for determining the sign of the fraction. Understanding these components will guide us to the solution.

Step-by-Step Solution

Alright, let's get down to the nitty-gritty and solve this inequality. We'll follow a systematic approach to make sure we don't miss any steps. Here's the breakdown:

Step 1: Factor the Denominator

The first thing we want to do is factor the denominator, $x^2 - 36$. As we mentioned earlier, this is a difference of squares, and it factors nicely into $(x - 6)(x + 6)$. So, our inequality now looks like this:

$\frac{6x}{(x - 6)(x + 6)} \geq 0$

Factoring the denominator helps us identify the values of x that make the denominator zero, which we'll need in the next step.

Step 2: Find the 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 are the potential places where the expression can change its sign (from positive to negative or vice versa). To find the critical points, we set both the numerator and the denominator equal to zero and solve for x.

• Numerator: $6x = 0$ gives us $x = 0$.
• Denominator: $(x - 6)(x + 6) = 0$ gives us $x = 6$ and $x = -6$.

So, our critical points are $x = -6$, $x = 0$, and $x = 6$. These points divide the number line into four intervals, which we'll analyze in the next step.

Step 3: Create a Sign Chart

A sign chart is a visual tool that helps us determine the sign of the expression in each interval created by the critical points. We'll create a number line and mark our critical points on it. Then, we'll test a value from each interval in the original inequality to see if the expression is positive or negative.

Here's how we set up the sign chart:

1. Draw a number line and mark the critical points: -6, 0, and 6.
2. These points divide the number line into four intervals: $(-\infty, -6)$, $(-6, 0)$, $(0, 6)$, and $(6, \infty)$.
3. Choose a test value from each interval and plug it into the factored inequality $\frac{6x}{(x - 6)(x + 6)} \geq 0$.

Let's pick some test values:

• Interval $(-\infty, -6)$: Test value $x = -7$ $\frac{6(-7)}{(-7 - 6)(-7 + 6)} = \frac{-42}{(-13)(-1)} = \frac{-42}{13} < 0$
• Interval $(-6, 0)$: Test value $x = -1$ $\frac{6(-1)}{(-1 - 6)(-1 + 6)} = \frac{-6}{(-7)(5)} = \frac{-6}{-35} = \frac{6}{35} > 0$
• Interval $(0, 6)$: Test value $x = 1$ $\frac{6(1)}{(1 - 6)(1 + 6)} = \frac{6}{(-5)(7)} = \frac{6}{-35} < 0$
• Interval $(6, \infty)$: Test value $x = 7$ $\frac{6(7)}{(7 - 6)(7 + 6)} = \frac{42}{(1)(13)} = \frac{42}{13} > 0$

Now, we can fill in the sign chart with the signs we found for each interval:

Step 4: Determine the Solution

We're looking for the intervals where the expression $\frac{6x}{(x - 6)(x + 6)}$ is greater than or equal to zero. This means we want the intervals where the sign is positive or zero. From our sign chart, we see that the expression is positive in the intervals $(-6, 0)$ and $(6, \infty)$.

Now, we need to consider the critical points. The inequality is $\geq 0$, so we include the values where the expression is equal to zero. The numerator is zero when $x = 0$, so we include $x = 0$ in our solution. However, the denominator is zero when $x = -6$ and $x = 6$, so we must exclude these values because they make the expression undefined (division by zero).

Therefore, the solution to the inequality is the union of the intervals $(-6, 0]$ and $(6, \infty)$.

Writing the Solution in Interval Notation

Finally, let's write the solution in interval notation. Interval notation is a way of representing intervals on the number line using brackets and parentheses. Square brackets [ ] indicate that the endpoint is included in the interval, while parentheses ( ) indicate that the endpoint is not included.

So, the solution to the inequality $\frac{6x}{x^2 - 36} \geq 0$ in interval notation is:

(-6, 0] ∪ (6, ∞)

Key Takeaways

Let's recap the key steps we took to solve this inequality:

1. Factor the denominator: Factoring helps identify critical points.
2. Find the critical points: These are the values where the expression can change signs.
3. Create a sign chart: This helps visualize the sign of the expression in each interval.
4. Determine the solution: Identify the intervals where the inequality is satisfied.
5. Write the solution in interval notation: Express the solution clearly and concisely.

Solving inequalities can seem tricky at first, but with practice, you'll get the hang of it. The sign chart is your best friend in these situations, so make sure you understand how to create and use one effectively.

Practice Makes Perfect

The best way to master solving inequalities is to practice! Try solving similar problems on your own. You can change the numbers, add more factors, or even work with different types of inequalities (like quadratic inequalities). The more you practice, the more confident you'll become.

And that's a wrap, guys! I hope this comprehensive guide has helped you understand how to solve the inequality $\frac{6x}{x^2 - 36} \geq 0$. Keep practicing, and you'll be a pro in no time! If you have any questions, feel free to ask. Happy solving!