Solve The Inequality:$ X - 3 \quad \frac{1}{2} \geq \quad \frac{1}{2} $

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. They are used to describe the relationship between different quantities, and solving inequalities is an essential skill in mathematics. In this article, we will focus on solving the given inequality: $ x - 3 \quad \frac{1}{2} \geq \quad \frac{1}{2} $. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is $ x - 3 \quad \frac{1}{2} \geq \quad \frac{1}{2} $. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to simplify the left-hand side of the inequality by combining the terms.

Simplifying the Left-Hand Side

To simplify the left-hand side, we need to get rid of the fraction $\frac{1}{2}$. We can do this by multiplying both sides of the inequality by 2. This will eliminate the fraction and make it easier to work with.

2(x - 3 \quad \frac{1}{2}) \geq 2(\frac{1}{2})

Distributing the 2

Now that we have multiplied both sides by 2, we can distribute the 2 to the terms inside the parentheses.

2x - 6 \quad \frac{1}{2} \geq 1

Adding 6 to Both Sides

The next step is to add 6 to both sides of the inequality to isolate the term with the variable $x$.

2x - 6 \quad \frac{1}{2} + 6 \geq 1 + 6

Simplifying the Inequality

Now that we have added 6 to both sides, we can simplify the inequality by combining the terms.

2x - 6 + 6 \quad \frac{1}{2} \geq 7

Combining Like Terms

The next step is to combine the like terms on the left-hand side of the inequality.

2x \quad \frac{1}{2} \geq 7

Adding $\frac{1}{2}$ to Both Sides

The next step is to add $\frac{1}{2}$ to both sides of the inequality to isolate the term with the variable $x$.

2x \quad \frac{1}{2} + \frac{1}{2} \geq 7 + \frac{1}{2}

Simplifying the Inequality

Now that we have added $\frac{1}{2}$ to both sides, we can simplify the inequality by combining the terms.

2x \geq 7 + \frac{1}{2}

Converting the Mixed Number to an Improper Fraction

The next step is to convert the mixed number $7 + \frac{1}{2}$ to an improper fraction.

7 + \frac{1}{2} = \frac{15}{2}

Substituting the Improper Fraction

Now that we have converted the mixed number to an improper fraction, we can substitute it into the inequality.

2x \geq \frac{15}{2}

Dividing Both Sides by 2

The final step is to divide both sides of the inequality by 2 to isolate the variable $x$.

x \geq \frac{15}{4}

Conclusion

In this article, we have solved the inequality $ x - 3 \quad \frac{1}{2} \geq \quad \frac{1}{2} $. We have broken down the solution step by step and provided a clear explanation of each step. The final solution is $x \geq \frac{15}{4}$. This means that the value of $x$ must be greater than or equal to $\frac{15}{4}$.

Final Answer

The final answer is $\boxed{\frac{15}{4}}$.

Related Topics

• Solving Linear Inequalities
• Graphing Linear Inequalities
• Solving Quadratic Inequalities
• Graphing Quadratic Inequalities

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Precalculus" by Michael Sullivan

Note: The references provided are for general information purposes only and are not specific to the solution of the given inequality.

Introduction

In our previous article, we solved the inequality $ x - 3 \quad \frac{1}{2} \geq \quad \frac{1}{2} $. We broke down the solution step by step and provided a clear explanation of each step. In this article, we will answer some frequently asked questions related to the solution of the inequality.

Q&A

Q: What is the final solution of the inequality?

A: The final solution of the inequality is $x \geq \frac{15}{4}$.

Q: How do I simplify the left-hand side of the inequality?

A: To simplify the left-hand side, you need to get rid of the fraction $\frac{1}{2}$. You can do this by multiplying both sides of the inequality by 2.

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

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality.

Q: What is the relationship between the solution of an inequality and the graph of the corresponding equation?

A: The solution of an inequality is the set of all values of the variable that satisfy the inequality. The graph of the corresponding equation is a visual representation of the solution set.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to plug the value of the variable into the original inequality and see if it is true.

Conclusion

In this article, we have answered some frequently asked questions related to the solution of the inequality $ x - 3 \quad \frac{1}{2} \geq \quad \frac{1}{2} $. We have provided clear explanations and examples to help you understand the concepts. If you have any more questions, feel free to ask.

Final Answer

The final answer is $\boxed{\frac{15}{4}}$.

Related Topics

• Solving Linear Inequalities
• Graphing Linear Inequalities
• Solving Quadratic Inequalities
• Graphing Quadratic Inequalities

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Precalculus" by Michael Sullivan

Note: The references provided are for general information purposes only and are not specific to the solution of the given inequality.