Solve The Rational Inequality $\frac{3}{x-4} \geq 1$. Express The Answer In Interval Form.A. (4,7) B. (4,7] C. [4,7) D. [4,7]

Introduction

Rational inequalities are a type of mathematical expression that involves a rational function, which is a ratio of two polynomials. Solving rational inequalities requires a combination of algebraic and analytical techniques to find the values of the variable that satisfy the given inequality. In this article, we will focus on solving the rational inequality $\frac{3}{x-4} \geq 1$ and express the answer in interval form.

Understanding Rational Inequalities

A rational inequality is an expression of the form $\frac{f(x)}{g(x)} \geq k$, where $f(x)$ and $g(x)$ are polynomials and $k$ is a constant. The goal of solving a rational inequality is to find the values of $x$ that make the inequality true. This can be done by finding the values of $x$ that make the numerator and denominator of the rational function equal to zero, and then using these values to determine the intervals where the inequality is true.

Step 1: Write the Inequality in a More Manageable Form

To solve the rational inequality $\frac{3}{x-4} \geq 1$, we can start by subtracting $1$ from both sides of the inequality. This gives us:

$$\frac{3}{x-4} - 1 \geq 0$$

We can then simplify the left-hand side of the inequality by finding a common denominator:

$$\frac{3}{x-4} - \frac{x-4}{x-4} \geq 0$$

This simplifies to:

$$\frac{3 - (x-4)}{x-4} \geq 0$$

Simplifying further, we get:

$$\frac{7-x}{x-4} \geq 0$$

Step 2: Find the Critical Points

The critical points of a rational function are the values of $x$ that make the numerator and denominator equal to zero. In this case, the critical points are $x = 7$ and $x = 4$. These points divide the number line into three intervals: $(-\infty, 4)$, $(4, 7)$, and $(7, \infty)$.

Step 3: Test Each Interval

To determine which intervals satisfy the inequality, we can test each interval by plugging in a test value from each interval into the inequality. Let's start with the interval $(-\infty, 4)$.

If we plug in $x = 0$ into the inequality, we get:

$$\frac{7-0}{0-4} \geq 0$$

This simplifies to:

$$\frac{7}{-4} \geq 0$$

Since $\frac{7}{-4} < 0$, the interval $(-\infty, 4)$ does not satisfy the inequality.

Next, let's test the interval $(4, 7)$.

If we plug in $x = 5$ into the inequality, we get:

$$\frac{7-5}{5-4} \geq 0$$

This simplifies to:

$$\frac{2}{1} \geq 0$$

Since $2 > 0$, the interval $(4, 7)$ satisfies the inequality.

Finally, let's test the interval $(7, \infty)$.

If we plug in $x = 8$ into the inequality, we get:

$$\frac{7-8}{8-4} \geq 0$$

This simplifies to:

$$\frac{-1}{4} \geq 0$$

Since $-1 < 0$, the interval $(7, \infty)$ does not satisfy the inequality.

Conclusion

Based on our analysis, we can conclude that the interval $(4, 7)$ satisfies the rational inequality $\frac{3}{x-4} \geq 1$. Therefore, the correct answer is:

A. (4,7)

$$$$

Introduction

In our previous article, we explored the concept of rational inequalities and provided a step-by-step guide on how to solve the rational inequality $\frac{3}{x-4} \geq 1$. In this article, we will address some common questions and concerns that students may have when working with rational inequalities.

Q: What is the difference between a rational inequality and a rational equation?

A: A rational equation is an equation that involves a rational function, which is a ratio of two polynomials. A rational inequality, on the other hand, is an inequality that involves a rational function. While rational equations and inequalities share some similarities, they have distinct differences in terms of their solution sets and the techniques used to solve them.

Q: How do I determine the critical points of a rational function?

A: The critical points of a rational function are the values of $x$ that make the numerator and denominator equal to zero. To find the critical points, you can set the numerator and denominator equal to zero and solve for $x$. For example, in the rational function $\frac{3}{x-4}$, the critical points are $x = 7$ and $x = 4$.

Q: What is the significance of the sign chart in solving rational inequalities?

A: The sign chart is a graphical representation of the sign of the rational function in different intervals. By analyzing the sign chart, you can determine which intervals satisfy the inequality. The sign chart is a useful tool in solving rational inequalities because it allows you to visualize the behavior of the rational function and identify the intervals where the inequality is true.

Q: How do I test each interval in a rational inequality?

A: To test each interval, you can plug in a test value from each interval into the inequality. The test value should be a value that is not equal to the critical points. For example, if you are testing the interval $(4, 7)$, you can plug in $x = 5$ into the inequality. If the inequality is true for the test value, then the interval satisfies the inequality.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict symbol, such as $>$ or $<$. A non-strict inequality is an inequality that is written with a non-strict symbol, such as $\geq$ or $\leq$. In a strict inequality, the endpoints are not included in the solution set, while in a non-strict inequality, the endpoints are included.

Q: How do I express the solution set of a rational inequality in interval form?

A: To express the solution set in interval form, you need to identify the intervals that satisfy the inequality. You can do this by analyzing the sign chart and testing each interval. Once you have identified the intervals that satisfy the inequality, you can express the solution set in interval form using the following notation:

• $(a, b)$: the interval includes all values between $a$ and $b$, but excludes the endpoints.
• $[a, b)$: the interval includes all values between $a$ and $b$, including the endpoint $a$, but excludes the endpoint $b$.
• $(a, b]$ : the interval includes all values between $a$ and $b$, including the endpoint $b$, but excludes the endpoint $a$.
• $[a, b]$ : the interval includes all values between $a$ and $b$, including both endpoints.

Conclusion

In this article, we addressed some common questions and concerns that students may have when working with rational inequalities. We hope that this Q&A guide has provided you with a better understanding of rational inequalities and how to solve them. Remember to always analyze the sign chart and test each interval to determine the solution set of a rational inequality.