What Is The Solution Set To The Inequality $ (4x - 3)(2x - 1) \geq 0 $?A. $ \{x \mid X \leq 3 \text{ Or } X \geq 1\} $B. $ \{x \mid X \leq 2 \text{ Or } X \geq \frac{4}{3}\} $C. $ \{x \mid X \leq \frac{1}{2} \text{ Or } X

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Introduction

In mathematics, inequalities are a fundamental concept in algebra and are used to describe the relationship between two or more expressions. The solution set of an inequality is the set of all possible values of the variable that satisfy the inequality. In this article, we will explore the solution set to the inequality $(4x - 3)(2x - 1) \geq 0$.

Understanding the Inequality

To find the solution set, we need to understand the inequality and its components. The given inequality is a product of two expressions: $(4x - 3)$ and $(2x - 1)$. The product of two expressions is greater than or equal to zero if and only if both expressions have the same sign (either both positive or both negative).

Finding the Critical Points

To find the solution set, we need to find the critical points of the inequality. The critical points are the values of $x$ that make the expressions $(4x - 3)$ and $(2x - 1)$ equal to zero. We can find the critical points by setting each expression equal to zero and solving for $x$.

For the expression $(4x - 3)$, we set it equal to zero and solve for $x$:

$$4x - 3 = 0$$

$$4x = 3$$

$$x = \frac{3}{4}$$

For the expression $(2x - 1)$, we set it equal to zero and solve for $x$:

$$2x - 1 = 0$$

$$2x = 1$$

$$x = \frac{1}{2}$$

Creating a Sign Chart

To find the solution set, we can create a sign chart to determine the sign of each expression in different intervals. We will use the critical points we found earlier to divide the number line into intervals.

Interval	$(4x - 3)$	$(2x - 1)$	$(4x - 3)(2x - 1)$
$(-\infty, \frac{1}{2})$	$-$	$-$	$+$
$(\frac{1}{2}, \frac{3}{4})$	$-$	$+$	$-$
$(\frac{3}{4}, \infty)$	$+$	$+$	$+$

Analyzing the Sign Chart

From the sign chart, we can see that the product $(4x - 3)(2x - 1)$ is positive in the intervals $(-\infty, \frac{1}{2})$ and $(\frac{3}{4}, \infty)$. This means that the inequality $(4x - 3)(2x - 1) \geq 0$ is satisfied when $x$ is in these intervals.

Writing the Solution Set

Based on the analysis of the sign chart, we can write the solution set to the inequality $(4x - 3)(2x - 1) \geq 0$ as:

$$\{x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4}\}$$

Conclusion

In this article, we explored the solution set to the inequality $(4x - 3)(2x - 1) \geq 0$. We found the critical points of the inequality, created a sign chart to determine the sign of each expression in different intervals, and analyzed the sign chart to find the solution set. The solution set to the inequality is $\{x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4}\}$.

Comparison with Other Options

Let's compare our solution set with the other options:

A. $\{x \mid x \leq 3 \text{ or } x \geq 1\}$

This option is incorrect because it does not include the critical point $x = \frac{3}{4}$.

B. $\{x \mid x \leq 2 \text{ or } x \geq \frac{4}{3}\}$

This option is incorrect because it does not include the critical point $x = \frac{1}{2}$.

C. $\{x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{4}{3}\}$

This option is incorrect because it includes the critical point $x = \frac{4}{3}$, which is not part of the solution set.

Final Answer

The final answer is $\boxed{\{x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4}\}}$.

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Q: What is the solution set to the inequality $(4x - 3)(2x - 1) \geq 0$?

A: The solution set to the inequality $(4x - 3)(2x - 1) \geq 0$ is $\{x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4}\}$.

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

A: To find the critical points of the inequality, you need to set each expression $(4x - 3)$ and $(2x - 1)$ equal to zero and solve for $x$. The critical points are $x = \frac{3}{4}$ and $x = \frac{1}{2}$.

Q: What is a sign chart and how do I use it to find the solution set?

A: A sign chart is a table that shows the sign of each expression in different intervals. To use a sign chart, you need to divide the number line into intervals using the critical points and determine the sign of each expression in each interval. The solution set is the set of all values of $x$ that make the product of the expressions greater than or equal to zero.

Q: Why is the solution set $\{x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4}\}$?

A: The solution set is $\{x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4}\}$ because the product of the expressions $(4x - 3)$ and $(2x - 1)$ is positive in the intervals $(-\infty, \frac{1}{2})$ and $(\frac{3}{4}, \infty)$. This means that the inequality $(4x - 3)(2x - 1) \geq 0$ is satisfied when $x$ is in these intervals.

Q: How do I compare the solution set with other options?

A: To compare the solution set with other options, you need to check if the other options include the critical points and the intervals where the product of the expressions is positive. If the other options do not include the critical points or the intervals where the product of the expressions is positive, then they are incorrect.

Q: What is the final answer to the inequality $(4x - 3)(2x - 1) \geq 0$?

A: The final answer to the inequality $(4x - 3)(2x - 1) \geq 0$ is $\boxed{\{x \mid x \leq \frac{1}{2} \text{ or } x \geq \frac{3}{4}\}}$.

Q: Can I use other methods to find the solution set?

A: Yes, you can use other methods to find the solution set, such as graphing the expressions and finding the intervals where the product of the expressions is positive. However, the method of creating a sign chart is a simple and effective way to find the solution set.

Q: What is the importance of finding the solution set to the inequality?

A: Finding the solution set to the inequality is important because it helps you understand the relationship between the expressions and the variable. It also helps you to make predictions and conclusions about the behavior of the expressions.

Q: Can I apply the method of creating a sign chart to other inequalities?

A: Yes, you can apply the method of creating a sign chart to other inequalities. The method is a general technique that can be used to find the solution set to any inequality that can be written in the form of a product of expressions.

Q: What are some common mistakes to avoid when finding the solution set to an inequality?

A: Some common mistakes to avoid when finding the solution set to an inequality include:

• Not including the critical points in the solution set
• Not including the intervals where the product of the expressions is positive
• Not using a sign chart to determine the sign of each expression in different intervals
• Not checking the solution set against other options

By avoiding these common mistakes, you can ensure that you find the correct solution set to the inequality.