Solve The Inequality:${ \frac{x}{3} - \frac{x}{2} + \frac{1}{2} \ \textgreater \ 1 \geq \frac{-x}{6}, \quad X \in \mathbb{R} }$

Mar 10, 2025 by ADMIN 131 views

Solving the Inequality: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of inequalities and explore a specific problem that requires us to solve an inequality involving fractions and variables. The given inequality is $\frac{x}{3} - \frac{x}{2} + \frac{1}{2} \ \textgreater \ 1 \geq \frac{-x}{6}$ , where $x$ is a real number. Our goal is to isolate the variable $x$ and determine the range of values that satisfy the given inequality.

Understanding the Inequality

Before we begin solving the inequality, let's break it down and understand what it's asking for. The given inequality is a compound inequality, which means it consists of two parts: $\frac{x}{3} - \frac{x}{2} + \frac{1}{2} \ \textgreater \ 1$ and $1 \geq \frac{-x}{6}$ . We need to find the values of $x$ that satisfy both parts of the inequality.

Step 1: Simplify the First Part of the Inequality

To simplify the first part of the inequality, we can start by combining the fractions on the left-hand side. We can do this by finding a common denominator, which is 6 in this case.

$\frac{x}{3} - \frac{x}{2} + \frac{1}{2} \ \textgreater \ 1$

We can rewrite the fractions with a common denominator:

$\frac{2x}{6} - \frac{3x}{6} + \frac{3}{6} \ \textgreater \ 1$

Now, we can combine the fractions:

$\frac{-x + 3}{6} \ \textgreater \ 1$

Step 2: Multiply Both Sides by 6

To get rid of the fraction, we can multiply both sides of the inequality by 6.

$-x + 3 \ \textgreater \ 6$

Step 3: Subtract 3 from Both Sides

Next, we can subtract 3 from both sides of the inequality to isolate the variable $x$ .

$-x \ \textgreater \ 3$

Step 4: Multiply Both Sides by -1

To make the inequality easier to read, we can multiply both sides by -1. Remember that when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality.

$x \ \textless \ -3$

Step 5: Solve the Second Part of the Inequality

Now, let's move on to the second part of the inequality: $1 \geq \frac{-x}{6}$ . We can start by multiplying both sides of the inequality by 6 to get rid of the fraction.

$6 \geq -x$

Step 6: Multiply Both Sides by -1

$-6 \leq x$

Step 7: Combine the Results

Now that we have solved both parts of the inequality, we can combine the results to find the range of values that satisfy the given inequality.

$x \ \textless \ -3$ and $-6 \leq x$

Conclusion

In this article, we have solved the inequality $\frac{x}{3} - \frac{x}{2} + \frac{1}{2} \ \textgreater \ 1 \geq \frac{-x}{6}$ , where $x$ is a real number. We have broken down the inequality into two parts and solved each part separately. The final result is that the range of values that satisfy the given inequality is $-6 \leq x \ \textless \ -3$ .

Final Answer

The final answer is $\boxed{-6 \leq x \ \textless \ -3}$ .

Discussion

The given inequality is a compound inequality, which means it consists of two parts. We need to find the values of $x$ that satisfy both parts of the inequality. The first part of the inequality is $\frac{x}{3} - \frac{x}{2} + \frac{1}{2} \ \textgreater \ 1$ , and the second part is $1 \geq \frac{-x}{6}$ . We have solved both parts of the inequality separately and combined the results to find the range of values that satisfy the given inequality.

References

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Introduction

In our previous article, we solved the inequality $\frac{x}{3} - \frac{x}{2} + \frac{1}{2} \ \textgreater \ 1 \geq \frac{-x}{6}$ , where $x$ is a real number. We broke down the inequality into two parts and solved each part separately. In this article, we will answer some frequently asked questions about solving inequalities and provide additional guidance on how to approach these types of problems.

Q&A

Q: What is an inequality?

A: An inequality is a statement that compares two expressions using a relation such as greater than, less than, greater than or equal to, or less than or equal to.

Q: What is a compound inequality?

A: A compound inequality is an inequality that consists of two or more parts, separated by the word "and" or "or".

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can follow these steps:

Simplify the inequality by combining like terms.
Isolate the variable by adding or subtracting the same value from both sides of the inequality.
Multiply or divide both sides of the inequality by a positive or negative number, as needed.
Write the solution in interval notation.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you can follow these steps:

Break down the compound inequality into two or more separate inequalities.
Solve each separate inequality using the steps outlined above.
Combine the solutions to the separate inequalities to find the final solution.

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 $x > 2$ . A quadratic inequality is an inequality that involves a quadratic expression, such as $x^2 > 4$ .

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you can follow these steps:

Graph the related linear equation by plotting two points and drawing a line through them.
Test a point in each region of the graph to determine which region satisfies the inequality.
Shade the region that satisfies the inequality.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you can follow these steps:

Graph each separate inequality using the steps outlined above.
Combine the graphs of the separate inequalities to find the final graph.

Additional Tips and Resources

When solving inequalities, it's essential to follow the order of operations (PEMDAS) and to simplify the inequality as much as possible.
When graphing inequalities, it's essential to test a point in each region of the graph to determine which region satisfies the inequality.
For additional guidance on solving inequalities, you can consult a textbook or online resource, such as Khan Academy or Mathway.

Conclusion

In this article, we have answered some frequently asked questions about solving inequalities and provided additional guidance on how to approach these types of problems. We have also discussed the importance of following the order of operations and testing points in each region of the graph when solving and graphing inequalities. By following these tips and resources, you can become more confident and proficient in solving inequalities.

Final Answer

The final answer is $\boxed{-6 \leq x \ \textless \ -3}$ .

Solve The Inequality:${ \frac{x}{3} - \frac{x}{2} + \frac{1}{2} \ \textgreater \ 1 \geq \frac{-x}{6}, \quad X \in \mathbb{R} }$

Introduction

Understanding the Inequality

Step 1: Simplify the First Part of the Inequality

Step 2: Multiply Both Sides by 6

Step 3: Subtract 3 from Both Sides

Step 4: Multiply Both Sides by -1

Step 5: Solve the Second Part of the Inequality

Step 6: Multiply Both Sides by -1

Step 7: Combine the Results

Conclusion

Final Answer

Discussion

Related Topics

References

Introduction

Q&A

Q: What is an inequality?

Q: What is a compound inequality?

Q: How do I solve a linear inequality?

Q: How do I solve a compound inequality?

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

Q: How do I graph a linear inequality?

Q: How do I graph a compound inequality?

Additional Tips and Resources

Conclusion

Final Answer

Discussion

Related Topics

References