Select All The Correct Answers.Which Values Are In The Solution Set To This Inequality?${ \frac{3x}{4} \ \textless \ 9 \quad \text{and} \quad -\frac{x}{6} \geq -3 }$A. { X = 18$}$B. { X = 15$}$C. { X = 11$}$D.

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. In this article, we will focus on solving a given inequality and determining the values that are in the solution set. We will use the given inequality $\frac{3x}{4} \ \textless \ 9 \quad \text{and} \quad -\frac{x}{6} \geq -3$ and find the correct values that satisfy this inequality.

Understanding the Inequality

The given inequality consists of two parts: $\frac{3x}{4} \ \textless \ 9$ and $-\frac{x}{6} \geq -3$. To solve this inequality, we need to isolate the variable $x$ in each part.

Part 1: $\frac{3x}{4} \ \textless \ 9$

To isolate $x$, we can start by multiplying both sides of the inequality by 4, which is the denominator of the fraction. This gives us:

$$3x \ \textless \ 36$$

Next, we can divide both sides of the inequality by 3 to get:

$$x \ \textless \ 12$$

Part 2: $-\frac{x}{6} \geq -3$

To isolate $x$, we can start by multiplying both sides of the inequality by -6, which is the denominator of the fraction. This gives us:

$$x \leq 18$$

Note that when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

Combining the Inequalities

Now that we have solved each part of the inequality, we can combine them to find the solution set. We need to find the values of $x$ that satisfy both inequalities.

From Part 1, we have $x \ \textless \ 12$, and from Part 2, we have $x \leq 18$. To find the solution set, we need to find the values of $x$ that satisfy both inequalities.

Since $x \ \textless \ 12$ is a strict inequality, we know that $x$ cannot be equal to 12. Therefore, the solution set is $x \ \textless \ 12$ and $x \leq 18$.

Finding the Correct Answers

Now that we have found the solution set, we can determine which of the given answers are correct.

A. $x = 18$ is not in the solution set because $x \ \textless \ 12$.

B. $x = 15$ is in the solution set because $15 \ \textless \ 12$ and $15 \leq 18$.

C. $x = 11$ is in the solution set because $11 \ \textless \ 12$ and $11 \leq 18$.

D. $x = 20$ is not in the solution set because $20 \ \textgreater \ 12$.

Therefore, the correct answers are B and C.

Conclusion

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Introduction

In our previous article, we solved a given inequality and determined the values that are in the solution set. In this article, we will provide a Q&A guide to help you understand the concept of solving inequalities and how to apply it to different problems.

Q: What is an inequality?

A: An inequality is a statement that compares two expressions using a mathematical symbol, such as <, >, ≤, or ≥.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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

A: A strict inequality is an inequality that uses a symbol such as < or >, while a non-strict inequality is an inequality that uses a symbol such as ≤ or ≥. For example, the inequality x < 5 is a strict inequality, while the inequality x ≤ 5 is a non-strict inequality.

Q: How do I combine two inequalities?

A: To combine two inequalities, you need to find the values of the variable that satisfy both inequalities. You can do this by finding the intersection of the two solution sets.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values of the variable that satisfy the inequality.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to plot a point on the number line that represents the value of the variable, and then shade the region to the left or right of the point, depending on the direction of the inequality sign.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

• Linear inequalities: inequalities that can be written in the form ax + b < c or ax + b > c
• Quadratic inequalities: inequalities that can be written in the form ax^2 + bx + c < 0 or ax^2 + bx + c > 0
• Rational inequalities: inequalities that can be written in the form a/x < b or a/x > b

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the factored form to find the solution set.

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

A: A rational inequality is an inequality that involves a rational expression, while a rational equation is an equation that involves a rational expression.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the values of the variable that make the rational expression equal to zero, and then use the resulting equation to find the solution set.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of solving inequalities and how to apply it to different problems. We covered topics such as solving linear inequalities, quadratic inequalities, and rational inequalities, and provided examples and explanations to help you understand the concepts.