Solving Rational Inequalities A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of rational inequalities. You know, those problems that look a little intimidating with fractions and variables all mixed together? Don't worry, we're going to break it down step by step, making it super easy to understand. We'll tackle a specific example, but the principles we learn here will help you solve almost any rational inequality you come across. So, buckle up and let's get started!

Understanding Rational Inequalities

Let's start with the basics. A rational inequality is simply an inequality that involves rational expressions. Remember, a rational expression is just a fraction where the numerator and denominator are polynomials. These inequalities often involve comparing a rational expression to zero, which is a key part of how we solve them. Our main goal is to find the values of the variable (usually 'x') that make the inequality true. This often involves a bit of algebraic manipulation and some careful consideration of the expression's behavior.

The core idea behind solving rational inequalities is identifying the critical points, which are the values of x that make either the numerator or the denominator of the rational expression equal to zero. These critical points divide the number line into intervals, and the sign of the rational expression within each interval remains constant. By testing a value from each interval, we can determine where the inequality holds true. It’s like creating a map of where the expression is positive, negative, or zero, helping us pinpoint the solutions. This method ensures we don't miss any potential solutions and accurately represent the inequality's solution set.

When dealing with rational inequalities, it's essential to remember that the denominator cannot be zero. This means any value of x that makes the denominator zero is excluded from the solution set. These values are critical points where the expression is undefined, creating vertical asymptotes on the graph of the rational function. Therefore, identifying these points and excluding them from the solution is a crucial step in solving rational inequalities. Ignoring this can lead to incorrect solutions and a misunderstanding of the inequality’s behavior. By carefully considering the domain of the rational expression, we ensure our solutions are accurate and meaningful.

Example Problem: $\frac{x^2-49}{x2-25}

Okay, let's jump into our example problem: $\frac{x^2-49}{x2-25}

Step 1: Find the Critical Points

The very first thing we need to do is find those crucial critical points. These are the values of x that make the numerator or the denominator equal to zero. Why? Because these points are where the expression can change its sign (from positive to negative or vice versa). They're like the dividing lines on our number line that separate the intervals we need to test.

So, let's start with the numerator, $x^2 - 49$. We need to solve the equation $x^2 - 49 = 0$. This is a classic difference of squares, which we can factor as $(x - 7)(x + 7) = 0$. Setting each factor to zero gives us $x = 7$ and $x = -7$. These are two of our critical points.

Now, let's tackle the denominator, $x^2 - 25$. We need to solve $x^2 - 25 = 0$. Again, this is a difference of squares, factoring as $(x - 5)(x + 5) = 0$. Setting each factor to zero gives us $x = 5$ and $x = -5$. These are our other two critical points.

So, in total, we have four critical points: -7, -5, 5, and 7. These are the key values we'll use to divide our number line into intervals.

Step 2: Create a Sign Chart

Alright, now we get to the fun part: creating a sign chart! This is a super helpful tool for visualizing how the expression's sign changes across different intervals. Think of it like a map that shows us where the expression is positive, negative, or zero.

First, we draw a number line and mark our critical points (-7, -5, 5, and 7) on it. These points divide the number line into five intervals: $(-\infty, -7)$, $(-7, -5)$, $(-5, 5)$, $(5, 7)$, and $(7, \infty)$.

Next, we need to figure out the sign of the expression $\frac{x^2-49}{x^2-25}$ in each interval. To do this, we'll pick a test value within each interval and plug it into the expression. We don't need to calculate the exact value; we just care about the sign (positive or negative).

Here’s how we can structure our sign chart:

Let's walk through a couple of these to make sure we're on the same page. For the interval $(-\infty, -7)$, we picked a test value of -8. Plugging -8 into $x^2 - 49$ gives us $(-8)^2 - 49 = 64 - 49 = 15$, which is positive. Plugging -8 into $x^2 - 25$ gives us $(-8)^2 - 25 = 64 - 25 = 39$, which is also positive. So, the entire expression $\frac{x^2-49}{x^2-25}$ is positive in this interval (positive divided by positive is positive).

For the interval $(-7, -5)$, we picked a test value of -6. Plugging -6 into $x^2 - 49$ gives us $(-6)^2 - 49 = 36 - 49 = -13$, which is negative. Plugging -6 into $x^2 - 25$ gives us $(-6)^2 - 25 = 36 - 25 = 11$, which is positive. So, the expression $\frac{x^2-49}{x^2-25}$ is negative in this interval (negative divided by positive is negative).

We repeat this process for each interval, and the sign chart gives us a clear picture of where the expression is positive, negative, or zero.

Step 3: Determine the Solution

Okay, we've done the hard work of finding the critical points and creating the sign chart. Now comes the moment of truth: figuring out the solution to our inequality, $\frac{x^2-49}{x^2-25} \leq 0$.

Remember, we're looking for the values of x that make the expression less than or equal to zero. This means we're interested in the intervals where the expression is negative or zero. Looking at our sign chart, we see that the expression is negative in the intervals $(-7, -5)$ and $(5, 7)$.

But what about the critical points themselves? We need to be careful here. The inequality is $\leq 0$, which means