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. \[$x = 12\$\] B. \[$x = 15\$\] C. \[$x =

Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving a system of linear inequalities and determining the values in the solution set.

Understanding the Inequality

The given inequality is a system of two linear inequalities:

{ \frac{3x}{4} \ \textless \ 9 \quad \text{and} \quad -\frac{x}{6} \geq -3 \}

To solve this system, we need to find the values of $x$ that satisfy both inequalities.

Solving the First Inequality

The first inequality is $\frac{3x}{4} \ \textless \ 9$. To solve for $x$, we can multiply both sides of the inequality by $\frac{4}{3}$.

{ \frac{3x}{4} \ \textless \ 9 \Rightarrow \frac{4}{3} \cdot \frac{3x}{4} \ \textless \ \frac{4}{3} \cdot 9 \}

Simplifying the inequality, we get:

{ x \ \textless \ 12 \}

This means that the solution set for the first inequality is all values of $x$ that are less than 12.

Solving the Second Inequality

The second inequality is $-\frac{x}{6} \geq -3$. To solve for $x$, we can multiply both sides of the inequality by $-6$.

{ -\frac{x}{6} \geq -3 \Rightarrow -6 \cdot -\frac{x}{6} \geq -6 \cdot -3 \}

Simplifying the inequality, we get:

{ x \leq 18 \}

This means that the solution set for the second inequality is all values of $x$ that are less than or equal to 18.

Finding the Intersection of the Solution Sets

To find the values in the solution set, we need to find the intersection of the solution sets of the two inequalities. The intersection of two sets is the set of elements that are common to both sets.

In this case, the solution set of the first inequality is all values of $x$ that are less than 12, and the solution set of the second inequality is all values of $x$ that are less than or equal to 18.

The intersection of these two solution sets is the set of values that are common to both sets, which is all values of $x$ that are less than 12 and less than or equal to 18.

Determining the Correct Answers

Based on the solution set, we can determine which of the given answers are correct.

A. $x = 12$ is not in the solution set because it is not less than 12.

B. $x = 15$ is in the solution set because it is less than 12 and less than or equal to 18.

C. $x = 18$ is not in the solution set because it is not less than 12.

Therefore, the correct answer is B. $x = 15$.

Conclusion

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Introduction

In our previous article, we solved a system of linear inequalities and determined the values in the solution set. In this article, we will provide a Q&A guide to help you understand the concepts and solve similar problems.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using greater than, less than, greater than or equal to, or less than or equal to.

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 linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

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 sign of the quadratic expression to determine the solution set.

Q: What is the solution set of an inequality?

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

Q: How do I find the solution set of a system of linear inequalities?

A: To find the solution set of a system of linear inequalities, you need to find the intersection of the solution sets of each inequality.

Q: What is the intersection of two sets?

A: The intersection of two sets is the set of elements that are common to both sets.

Q: How do I determine the correct answer to a problem?

A: To determine the correct answer to a problem, you need to check if the answer satisfies the inequality.

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

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

• Not isolating the variable on one side of the inequality sign
• Not checking if the answer satisfies the inequality
• Not considering the sign of the inequality

Conclusion

Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we provided a Q&A guide to help you understand the concepts and solve similar problems. We also discussed common mistakes to avoid when solving inequalities. By following these tips and practicing with examples, you can become proficient in solving inequalities and apply them to real-world problems.

Frequently Asked Questions

• Q: What is the difference between a linear inequality and a quadratic inequality? A: A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.
• 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 sign of the quadratic expression to determine the solution set.
• Q: What is the solution set of an inequality? A: The solution set of an inequality is the set of values that satisfy the inequality.
• Q: How do I find the solution set of a system of linear inequalities? A: To find the solution set of a system of linear inequalities, you need to find the intersection of the solution sets of each inequality.

Additional Resources

• Inequality Solver: A online tool that can help you solve inequalities.
• Inequality Calculator: A online tool that can help you calculate the solution set of an inequality.
• Inequality Tutorial: A tutorial that provides a step-by-step guide to solving inequalities.