Solve The Inequality:$ -\frac{2}{11} J \leq 8 }$Possible Solutions A. { J \leq -44 $ $B. { J \geq -44 $}$C. { J \leq 8 \frac{2}{11} $}$

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving a specific type of inequality, which is a linear inequality. We will use the given inequality $-\frac{2}{11} J \leq 8$ as an example to demonstrate the step-by-step process of solving it.

Understanding the Inequality

Before we dive into solving the inequality, let's first understand what it means. The given inequality is $-\frac{2}{11} J \leq 8$. This means that the expression $-\frac{2}{11} J$ is less than or equal to 8. In other words, the value of $J$ must be such that when multiplied by $-\frac{2}{11}$, the result is less than or equal to 8.

Step 1: Isolate the Variable

To solve the inequality, we need to isolate the variable $J$. This means that we need to get $J$ by itself on one side of the inequality. To do this, we can start by multiplying both sides of the inequality by $-\frac{11}{2}$, which is the reciprocal of $-\frac{2}{11}$. This will cancel out the $-\frac{2}{11}$ term on the left-hand side of the inequality.

-\frac{2}{11} J \leq 8
\Rightarrow J \geq -\frac{11}{2} \cdot 8
\Rightarrow J \geq -44

Step 2: Write the Solution in Interval Notation

Now that we have isolated the variable $J$, we can write the solution in interval notation. The solution is $J \geq -44$, which means that $J$ can take on any value greater than or equal to $-44$.

Conclusion

In conclusion, we have solved the inequality $-\frac{2}{11} J \leq 8$ by isolating the variable $J$ and writing the solution in interval notation. The solution is $J \geq -44$, which means that $J$ can take on any value greater than or equal to $-44$.

Possible Solutions

Based on the solution we obtained, we can now determine the possible solutions to the inequality.

• A. $j \leq -44$: This is not a possible solution, as the solution we obtained is $J \geq -44$, not $J \leq -44$.
• B. $j \geq -44$: This is a possible solution, as it matches the solution we obtained.
• C. $j \leq 8 \frac{2}{11}$: This is not a possible solution, as the solution we obtained is $J \geq -44$, not $J \leq 8 \frac{2}{11}$.

Discussion

In this article, we have demonstrated the step-by-step process of solving a linear inequality. We used the given inequality $-\frac{2}{11} J \leq 8$ as an example and obtained the solution $J \geq -44$. We also discussed the possible solutions to the inequality and determined that only option B is a possible solution.

Solving Inequalities: Tips and Tricks

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Here are some tips and tricks to help you solve inequalities:

• Read the inequality carefully: Before you start solving the inequality, make sure you read it carefully and understand what it means.
• Isolate the variable: To solve the inequality, you need to isolate the variable. This means that you need to get the variable by itself on one side of the inequality.
• Use inverse operations: To isolate the variable, you may need to use inverse operations, such as multiplication and division.
• Check your solution: Once you have obtained the solution, make sure to check it by plugging it back into the original inequality.

By following these tips and tricks, you can become proficient in solving inequalities and tackle even the most challenging problems with confidence.

Conclusion

Introduction

In our previous article, we discussed the step-by-step process of solving linear inequalities. In this article, we will provide a Q&A guide to help you better understand the concept of solving inequalities. We will cover common questions and concerns that students often have when it comes to solving inequalities.

Q: What is an inequality?

A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. For example, the inequality $2x + 3 > 5$ means that the expression $2x + 3$ is greater than 5.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable. This means that you need to get the variable by itself on one side of the inequality. You can do this by using inverse operations, such as multiplication and division.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that involves a linear expression, such as $2x + 3 > 5$. A quadratic inequality, on the other hand, is an inequality that involves a quadratic expression, such as $x^2 + 4x + 4 > 0$.

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 expression to determine the solution. For example, if you have the inequality $x^2 + 4x + 4 > 0$, you can factor the expression as $(x + 2)^2 > 0$. Since the square of any real number is always non-negative, the solution to this inequality is $x > -2$.

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

A: A strict inequality is an inequality that uses the symbols $<$ or $>$, such as $2x + 3 > 5$. A non-strict inequality, on the other hand, is an inequality that uses the symbols $\leq$ or $\geq$, such as $2x + 3 \leq 5$.

Q: How do I determine the solution to a strict inequality?

A: To determine the solution to a strict inequality, you need to isolate the variable and then use the sign of the expression to determine the solution. For example, if you have the inequality $2x + 3 > 5$, you can isolate the variable by subtracting 3 from both sides and then dividing both sides by 2. This gives you the solution $x > 1$.

Q: How do I determine the solution to a non-strict inequality?

A: To determine the solution to a non-strict inequality, you need to isolate the variable and then use the sign of the expression to determine the solution. For example, if you have the inequality $2x + 3 \leq 5$, you can isolate the variable by subtracting 3 from both sides and then dividing both sides by 2. This gives you the solution $x \leq 1$.

Q: What is the importance of solving inequalities?

A: Solving inequalities is an essential skill in mathematics that has many real-world applications. Inequalities are used to model real-world problems, such as budgeting, finance, and science. By solving inequalities, you can make informed decisions and solve problems that involve uncertainty.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics that requires practice and patience. By following the step-by-step process outlined in this article, you can solve linear and quadratic inequalities with ease. Remember to read the inequality carefully, isolate the variable, use inverse operations, and check your solution. With practice and patience, you can become proficient in solving inequalities and tackle even the most challenging problems with confidence.

Frequently Asked Questions

• Q: What is the difference between a linear inequality and a quadratic inequality? A: A linear inequality is an inequality that involves a linear expression, such as $2x + 3 > 5$. A quadratic inequality, on the other hand, is an inequality that involves a quadratic expression, such as $x^2 + 4x + 4 > 0$.
• 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 expression to determine the solution.
• Q: What is the difference between a strict inequality and a non-strict inequality? A: A strict inequality is an inequality that uses the symbols $<$ or $>$, such as $2x + 3 > 5$. A non-strict inequality, on the other hand, is an inequality that uses the symbols $\leq$ or $\geq$, such as $2x + 3 \leq 5$.
• Q: How do I determine the solution to a strict inequality? A: To determine the solution to a strict inequality, you need to isolate the variable and then use the sign of the expression to determine the solution.
• Q: How do I determine the solution to a non-strict inequality? A: To determine the solution to a non-strict inequality, you need to isolate the variable and then use the sign of the expression to determine the solution.

Additional Resources

• Online Resources: There are many online resources available that can help you learn how to solve inequalities, including video tutorials, practice problems, and interactive quizzes.
• Textbooks: There are many textbooks available that cover the topic of solving inequalities, including algebra textbooks and mathematics textbooks.
• Tutors: If you are struggling to learn how to solve inequalities, consider hiring a tutor who can provide one-on-one instruction and support.

By following the tips and resources outlined in this article, you can become proficient in solving inequalities and tackle even the most challenging problems with confidence.