Solve The Inequality $\frac{3x-4}{-7} \geq 0$. Write The Solution Set Using Interval Notation And Graph It.Write The Solution Set Using Interval Notation. Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete

Introduction

In this article, we will focus on solving the inequality $\frac{3x-4}{-7} \geq 0$. We will break down the solution process into manageable steps, and then express the solution set using interval notation. Additionally, we will graph the solution set to visualize the results.

Step 1: Understand the Inequality

The given inequality is $\frac{3x-4}{-7} \geq 0$. To solve this inequality, we need to find the values of $x$ that make the expression $\frac{3x-4}{-7}$ greater than or equal to zero.

Step 2: Find the Critical Points

To find the critical points, we need to set the numerator and denominator equal to zero and solve for $x$. The numerator is $3x-4$, and the denominator is $-7$. Setting the numerator equal to zero, we get:

$$3x-4=0 \Rightarrow 3x=4 \Rightarrow x=\frac{4}{3}$$

Setting the denominator equal to zero, we get:

$$-7=0$$

This is not possible, so we only have one critical point, which is $x=\frac{4}{3}$.

Step 3: Test the Intervals

To test the intervals, we need to choose a test point from each interval and plug it into the inequality. The intervals are:

• $(-\infty, \frac{4}{3})$
• $(\frac{4}{3}, \infty)$

Let's choose a test point from each interval. For the first interval, let's choose $x=-1$. Plugging this value into the inequality, we get:

$$\frac{3(-1)-4}{-7}=\frac{-3-4}{-7}=\frac{-7}{-7}=1$$

Since $1$ is greater than zero, the inequality is true for $x=-1$. Therefore, the inequality is true for all values of $x$ in the interval $(-\infty, \frac{4}{3})$.

For the second interval, let's choose $x=2$. Plugging this value into the inequality, we get:

$$\frac{3(2)-4}{-7}=\frac{6-4}{-7}=\frac{2}{-7}=-\frac{2}{7}$$

Since $-\frac{2}{7}$ is less than zero, the inequality is false for $x=2$. Therefore, the inequality is false for all values of $x$ in the interval $(\frac{4}{3}, \infty)$.

Step 4: Write the Solution Set

Based on the results of the interval testing, we can write the solution set as:

$$x \in \boxed{(-\infty, \frac{4}{3})}$$

Graphing the Solution Set

To graph the solution set, we can use a number line. The solution set is the interval $(-\infty, \frac{4}{3})$, which includes all values of $x$ less than $\frac{4}{3}$. We can graph this interval by drawing a line on the number line and shading the region to the left of the line.

Conclusion

Introduction

In our previous article, we solved the inequality $\frac{3x-4}{-7} \geq 0$ using a step-by-step approach. We found the critical points, tested the intervals, and wrote the solution set using interval notation. We also graphed the solution set to visualize the results. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to understand the inequality and identify the critical points. Critical points are the values of the variable that make the expression on one side of the inequality equal to zero.

Q: How do I find the critical points of an inequality?

A: To find the critical points of an inequality, you need to set the numerator and denominator equal to zero and solve for the variable. The numerator is the expression on the left side of the inequality, and the denominator is the expression on the right side of the inequality.

Q: What is the difference between a critical point and a solution to an inequality?

A: A critical point is a value of the variable that makes the expression on one side of the inequality equal to zero. A solution to an inequality is a value of the variable that makes the inequality true.

Q: How do I test the intervals of an inequality?

A: To test the intervals of an inequality, you need to choose a test point from each interval and plug it into the inequality. If the inequality is true for the test point, then the inequality is true for all values of the variable in that interval.

Q: What is the purpose of graphing the solution set of an inequality?

A: The purpose of graphing the solution set of an inequality is to visualize the results and understand the relationship between the variable and the inequality.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, it's always a good idea to check your work by hand to make sure you understand the solution.

Q: How do I write the solution set of an inequality in interval notation?

A: To write the solution set of an inequality in interval notation, you need to use the following notation:

• $(-\infty, a)$ to represent all values of the variable less than $a$
• $(a, \infty)$ to represent all values of the variable greater than $a$
• $(-\infty, a]$ to represent all values of the variable less than or equal to $a$
• $[a, \infty)$ to represent all values of the variable greater than or equal to $a$

Q: Can I have multiple solution sets for an inequality?

A: Yes, you can have multiple solution sets for an inequality. This occurs when the inequality has multiple critical points or when the solution set is a union of multiple intervals.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We covered topics such as finding critical points, testing intervals, graphing the solution set, and writing the solution set in interval notation. We also discussed the use of calculators and the possibility of having multiple solution sets for an inequality. By understanding these concepts, you will be better equipped to solve inequalities and understand the relationships between variables and inequalities.