Which Choice Is The Solution To The Inequality Below? 9 X ≤ 54 9x \leq 54 9 X ≤ 54 A. X ≤ 45 X \leq 45 X ≤ 45 B. X ≥ 6 X \geq 6 X ≥ 6 C. X ≤ 3 X \leq 3 X ≤ 3 D. X ≤ 6 X \leq 6 X ≤ 6

Mar 13, 2025 by ADMIN 182 views

Solving Inequalities: A Step-by-Step Guide

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Introduction

In mathematics, inequalities are a fundamental concept that helps us compare values and solve problems. An inequality is a statement that two expressions are not equal, but one is either greater than, less than, or greater than or equal to the other. In this article, we will focus on solving linear inequalities, specifically the inequality $9x \leq 54$ . We will explore the different solution methods and provide step-by-step examples to help you understand the concept.

What is a Linear Inequality?

A linear inequality is an inequality that can be written in the form $ax \leq b$ , where $a$ and $b$ are constants, and $x$ is the variable. In this case, the inequality is $9x \leq 54$ . The goal is to isolate the variable $x$ and find the solution set that satisfies the inequality.

Solving the Inequality

To solve the inequality $9x \leq 54$ , we need to isolate the variable $x$ . We can do this by dividing both sides of the inequality by $9$ . This will give us the solution set that satisfies the inequality.

Step 1: Divide Both Sides by 9

When we divide both sides of the inequality by $9$ , we get:

\frac{9x}{9} \leq \frac{54}{9}

This simplifies to:

x \leq 6

Step 2: Write the Solution Set

The solution set is the set of all values of $x$ that satisfy the inequality. In this case, the solution set is $x \leq 6$ . This means that any value of $x$ that is less than or equal to $6$ is a solution to the inequality.

Analyzing the Answer Choices

Now that we have solved the inequality, let's analyze the answer choices:

A. $x \leq 45$ : This is not a solution to the inequality, as $x$ can be less than $45$ but still satisfy the inequality.
B. $x \geq 6$ : This is not a solution to the inequality, as $x$ can be greater than $6$ but still satisfy the inequality.
C. $x \leq 3$ : This is not a solution to the inequality, as $x$ can be less than $3$ but still satisfy the inequality.
D. $x \leq 6$ : This is the correct solution to the inequality.

Conclusion

Solving linear inequalities requires a step-by-step approach. We need to isolate the variable and find the solution set that satisfies the inequality. In this article, we solved the inequality $9x \leq 54$ and found the solution set to be $x \leq 6$ . We also analyzed the answer choices and found that only one of them was a correct solution to the inequality.

Frequently Asked Questions

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax \leq b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable and find the solution set that satisfies the inequality. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What is the solution set of a linear inequality?

A: The solution set is the set of all values of $x$ that satisfy the inequality. It is usually written in the form $x \leq a$ or $x \geq a$ , where $a$ is a constant.

Additional Resources

If you want to learn more about solving linear inequalities, here are some additional resources:

Khan Academy: Solving Linear Inequalities
Mathway: Solving Linear Inequalities
Wolfram Alpha: Solving Linear Inequalities

Final Thoughts

Solving linear inequalities is an essential skill in mathematics. It requires a step-by-step approach and a clear understanding of the concept. By following the steps outlined in this article, you can solve linear inequalities with confidence. Remember to always analyze the answer choices and choose the correct solution to the inequality.

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Introduction

In our previous article, we discussed how to solve linear inequalities, specifically the inequality $9x \leq 54$ . We provided step-by-step examples and analyzed the answer choices. In this article, we will continue to explore the concept of solving linear inequalities and provide a Q&A guide to help you understand the concept better.

Q&A Guide

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax \leq b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I solve a linear inequality?

Q: What is the solution set of a linear inequality?

A: The solution set is the set of all values of $x$ that satisfy the inequality. It is usually written in the form $x \leq a$ or $x \geq a$ , where $a$ is a constant.

Q: How do I know which direction to go when solving a linear inequality?

A: When solving a linear inequality, you need to determine the direction of the inequality. If the inequality is of the form $ax \leq b$ , you need to divide both sides by $a$ to isolate the variable. If the inequality is of the form $ax \geq b$ , you need to divide both sides by $a$ to isolate the variable.

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

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and find the values of $x$ that make the expression equal to zero. You can then use a number line or a graph to determine the solution set.

Q: What is the importance of solving linear inequalities?

A: Solving linear inequalities is an essential skill in mathematics. It is used in a variety of applications, including finance, economics, and science. It helps you to make informed decisions and solve real-world problems.

Real-World Applications

Solving linear inequalities has many real-world applications. Here are a few examples:

Finance: When investing in stocks or bonds, you need to determine the minimum or maximum return on investment. This can be done by solving a linear inequality.
Economics: When analyzing the demand for a product, you need to determine the minimum or maximum price that consumers are willing to pay. This can be done by solving a linear inequality.
Science: When conducting experiments, you need to determine the minimum or maximum value of a variable. This can be done by solving a linear inequality.

Conclusion

Frequently Asked Questions

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form $ax \leq b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I solve a linear inequality?

Q: What is the solution set of a linear inequality?

A: The solution set is the set of all values of $x$ that satisfy the inequality. It is usually written in the form $x \leq a$ or $x \geq a$ , where $a$ is a constant.

Additional Resources

If you want to learn more about solving linear inequalities, here are some additional resources:

Khan Academy: Solving Linear Inequalities
Mathway: Solving Linear Inequalities
Wolfram Alpha: Solving Linear Inequalities