What Is The Solution, If Any, To The Inequality ∣ 3 X ∣ ≥ U ? |3x| \geq U_{\text {? }} ∣3 X ∣ ≥ U ? ?A. All Real Numbers B. No Solution C. X ≥ 0 X \geq 0 X ≥ 0 D. X ≤ 0 X \leq 0 X ≤ 0

Mar 13, 2025 by ADMIN 188 views

$What is the Solution to the Inequality $|3x| \geq U_{\text {? }}$?$

Introduction

Inequalities are mathematical expressions that compare two values or expressions, often with a greater-than or less-than symbol. Absolute value inequalities, in particular, involve the absolute value function, which returns the distance of a number from zero on the number line. In this article, we will explore the solution to the inequality $|3x| \geq U_{\text {? }}$ , where $U_{\text {? }}$ is a value that we need to determine.

Understanding Absolute Value Inequalities

To solve absolute value inequalities, we need to consider two cases: when the expression inside the absolute value is positive, and when it is negative. This is because the absolute value function returns the distance of a number from zero, regardless of whether the number is positive or negative.

For the inequality $|3x| \geq U_{\text {? }}$ , we can start by considering the case when $3x$ is positive. In this case, the absolute value of $3x$ is equal to $3x$ itself, and the inequality becomes $3x \geq U_{\text {? }}$ . We can then divide both sides of the inequality by 3 to get $x \geq \frac{U_{\text {? }}}{3}$ .

Considering the Negative Case

Now, let's consider the case when $3x$ is negative. In this case, the absolute value of $3x$ is equal to $-3x$ , and the inequality becomes $-3x \geq U_{\text {? }}$ . We can then divide both sides of the inequality by -3, remembering to reverse the direction of the inequality sign, to get $x \leq -\frac{U_{\text {? }}}{3}$ .

Combining the Cases

To find the solution to the inequality $|3x| \geq U_{\text {? }}$ , we need to combine the two cases we considered earlier. This means that the solution is the union of the two intervals: $x \geq \frac{U_{\text {? }}}{3}$ and $x \leq -\frac{U_{\text {? }}}{3}$ .

Determining the Value of $U_{\text {? }}$

However, we are given that the solution is one of the following options: A. All real numbers, B. No solution, C. $x \geq 0$ , or D. $x \leq 0$ . To determine the value of $U_{\text {? }}$ , we need to consider each of these options in turn.

Option A: All Real Numbers

If the solution is all real numbers, then the inequality $|3x| \geq U_{\text {? }}$ must be true for all values of $x$ . This means that $U_{\text {? }}$ must be less than or equal to zero, since the absolute value of $3x$ is always non-negative.

Option B: No Solution

If there is no solution to the inequality $|3x| \geq U_{\text {? }}$ , then the inequality must be false for all values of $x$ . This means that $U_{\text {? }}$ must be greater than the absolute value of $3x$ for all values of $x$ .

Option C: $x \geq 0$

If the solution is $x \geq 0$ , then the inequality $|3x| \geq U_{\text {? }}$ must be true for all non-negative values of $x$ . This means that $U_{\text {? }}$ must be less than or equal to the absolute value of $3x$ for all non-negative values of $x$ .

Option D: $x \leq 0$

If the solution is $x \leq 0$ , then the inequality $|3x| \geq U_{\text {? }}$ must be true for all non-positive values of $x$ . This means that $U_{\text {? }}$ must be less than or equal to the absolute value of $3x$ for all non-positive values of $x$ .

Conclusion

In conclusion, the solution to the inequality $|3x| \geq U_{\text {? }}$ depends on the value of $U_{\text {? }}$ . If $U_{\text {? }}$ is less than or equal to zero, then the solution is all real numbers. If $U_{\text {? }}$ is greater than the absolute value of $3x$ for all values of $x$ , then there is no solution. If $U_{\text {? }}$ is less than or equal to the absolute value of $3x$ for all non-negative values of $x$ , then the solution is $x \geq 0$ . If $U_{\text {? }}$ is less than or equal to the absolute value of $3x$ for all non-positive values of $x$ , then the solution is $x \leq 0$ .

Final Answer

The final answer is: $\boxed{0}$

Q: What is the absolute value function?

A: The absolute value function, denoted by $|x|$ , returns the distance of a number $x$ from zero on the number line. It is defined as $|x| = x$ if $x \geq 0$ , and $|x| = -x$ if $x < 0$ .

Q: How do I solve absolute value inequalities?

A: To solve absolute value inequalities, you need to consider two cases: when the expression inside the absolute value is positive, and when it is negative. This is because the absolute value function returns the distance of a number from zero, regardless of whether the number is positive or negative.

Q: What is the solution to the inequality $|3x| \geq U_{\text {? }}$ ?

A: The solution to the inequality $|3x| \geq U_{\text {? }}$ depends on the value of $U_{\text {? }}$ . If $U_{\text {? }}$ is less than or equal to zero, then the solution is all real numbers. If $U_{\text {? }}$ is greater than the absolute value of $3x$ for all values of $x$ , then there is no solution. If $U_{\text {? }}$ is less than or equal to the absolute value of $3x$ for all non-negative values of $x$ , then the solution is $x \geq 0$ . If $U_{\text {? }}$ is less than or equal to the absolute value of $3x$ for all non-positive values of $x$ , then the solution is $x \leq 0$ .

Q: How do I determine the value of $U_{\text {? }}$ ?

A: To determine the value of $U_{\text {? }}$ , you need to consider each of the options: A. All real numbers, B. No solution, C. $x \geq 0$ , or D. $x \leq 0$ . You can do this by analyzing the inequality $|3x| \geq U_{\text {? }}$ and determining which option is consistent with the inequality.

Q: What is the significance of the absolute value of $3x$ ?

A: The absolute value of $3x$ represents the distance of $3x$ from zero on the number line. This is an important concept in solving absolute value inequalities, as it allows you to consider both the positive and negative cases.

Q: Can I use the same method to solve other absolute value inequalities?

A: Yes, you can use the same method to solve other absolute value inequalities. The key is to consider both the positive and negative cases, and to analyze the inequality to determine the solution.

Q: Are there any other types of inequalities that I should be aware of?

A: Yes, there are other types of inequalities that you should be aware of, such as linear inequalities, quadratic inequalities, and polynomial inequalities. Each of these types of inequalities has its own unique characteristics and solution methods.

Q: How can I practice solving absolute value inequalities?

A: You can practice solving absolute value inequalities by working through examples and exercises. You can also try solving inequalities with different coefficients and constants to see how the solution changes.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

Failing to consider both the positive and negative cases
Not analyzing the inequality carefully enough to determine the solution
Making errors in algebraic manipulations
Not checking the solution to ensure that it is consistent with the original inequality

Q: Can I use a calculator to solve absolute value inequalities?

A: Yes, you can use a calculator to solve absolute value inequalities. However, it's always a good idea to check the solution by hand to ensure that it is correct.

Q: Are there any online resources that I can use to learn more about absolute value inequalities?

A: Yes, there are many online resources that you can use to learn more about absolute value inequalities, including video tutorials, online textbooks, and practice problems. Some popular resources include Khan Academy, Mathway, and Wolfram Alpha.