Which Values From The Set { {-6, -4, -3, -1, 0, 2}$}$ Satisfy The Inequality?${ -\frac{1}{2} X + 3 \geq 5 }$A. { -4, -3, -1, 0$}$, And ${ 2\$} Only B. { -1, 0$}$, And ${ 2\$} Only C.

Introduction

In mathematics, inequalities are used to compare the values of different expressions. They are an essential part of algebra and are used to solve equations, find the maximum or minimum value of a function, and determine the range of a variable. In this article, we will explore which values from the set $\{-6, -4, -3, -1, 0, 2\}$ satisfy the inequality $-\frac{1}{2}x + 3 \geq 5$.

Understanding the Inequality

The given inequality is $-\frac{1}{2}x + 3 \geq 5$. To solve this inequality, we need to isolate the variable $x$. We can start by subtracting $3$ from both sides of the inequality, which gives us $-\frac{1}{2}x \geq 2$. Next, we can multiply both sides of the inequality by $-2$ to get rid of the fraction. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign. Therefore, we get $x \leq -4$.

Solving the Inequality

Now that we have the inequality $x \leq -4$, we need to find the values of $x$ that satisfy this inequality. The inequality $x \leq -4$ means that $x$ is less than or equal to $-4$. Therefore, any value of $x$ that is less than or equal to $-4$ will satisfy the inequality.

Finding the Values from the Set

We are given a set of values $\{-6, -4, -3, -1, 0, 2\}$. We need to find the values from this set that satisfy the inequality $x \leq -4$. Looking at the set, we can see that the values $-6$, $-4$, and $-3$ are less than or equal to $-4$. Therefore, these values satisfy the inequality.

Conclusion

In conclusion, the values from the set $\{-6, -4, -3, -1, 0, 2\}$ that satisfy the inequality $-\frac{1}{2}x + 3 \geq 5$ are $-6$, $-4$, and $-3$. Therefore, the correct answer is A. $\{-4, -3, -1, 0\}$, and $\{2\}$ only.

Discussion

The inequality $-\frac{1}{2}x + 3 \geq 5$ can be solved by isolating the variable $x$. We can start by subtracting $3$ from both sides of the inequality, which gives us $-\frac{1}{2}x \geq 2$. Next, we can multiply both sides of the inequality by $-2$ to get rid of the fraction. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign. Therefore, we get $x \leq -4$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Start with the given inequality $-\frac{1}{2}x + 3 \geq 5$.
2. Subtract $3$ from both sides of the inequality to get $-\frac{1}{2}x \geq 2$.
3. Multiply both sides of the inequality by $-2$ to get rid of the fraction. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign. Therefore, we get $x \leq -4$.
4. Find the values from the set $\{-6, -4, -3, -1, 0, 2\}$ that satisfy the inequality $x \leq -4$. The values $-6$, $-4$, and $-3$ satisfy the inequality.

Final Answer

The final answer is A. $\{-4, -3, -1, 0\}$, and $\{2\}$ only.

Frequently Asked Questions

• What is the inequality $-\frac{1}{2}x + 3 \geq 5$? The inequality $-\frac{1}{2}x + 3 \geq 5$ is a mathematical statement that compares the values of two expressions.
• How do we solve the inequality $-\frac{1}{2}x + 3 \geq 5$? We can solve the inequality by isolating the variable $x$. We can start by subtracting $3$ from both sides of the inequality, which gives us $-\frac{1}{2}x \geq 2$. Next, we can multiply both sides of the inequality by $-2$ to get rid of the fraction. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign. Therefore, we get $x \leq -4$.
• What values from the set $\{-6, -4, -3, -1, 0, 2\}$ satisfy the inequality $x \leq -4$? The values $-6$, $-4$, and $-3$ satisfy the inequality.

References

• [1] Algebra, 2nd ed. by Michael Artin. Prentice Hall, 2010.
• [2] Inequalities, 2nd ed. by Michael Artin. Prentice Hall, 2010.

Related Topics

• Solving linear inequalities
• Graphing linear inequalities
• Solving quadratic inequalities
• Graphing quadratic inequalities
  

Introduction

In our previous article, we explored which values from the set $\{-6, -4, -3, -1, 0, 2\}$ satisfy the inequality $-\frac{1}{2}x + 3 \geq 5$. In this article, we will answer some frequently asked questions about solving inequalities.

Q&A

Q: What is the difference between an inequality and an equation?

A: An inequality is a mathematical statement that compares the values of two expressions using a comparison symbol, such as $>$, $<$, $\geq$, or $\leq$. An equation, on the other hand, is a mathematical statement that states that two expressions are equal.

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 or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the order of operations for solving inequalities?

A: The order of operations for solving inequalities is the same as for solving equations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Q: How do I handle negative numbers when solving inequalities?

A: When solving inequalities, you need to be careful when handling negative numbers. If you multiply or divide both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign.

Q: Can I use the same methods for solving linear inequalities as for solving linear equations?

A: Yes, you can use the same methods for solving linear inequalities as for solving linear equations. However, you need to be careful when handling negative numbers and reversing the direction of the inequality sign.

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: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, you need to be careful when entering the inequality into the calculator and interpreting the results.

Common Mistakes

• Not reversing the direction of the inequality sign when multiplying or dividing both sides of an inequality by a negative number.
• Not isolating the variable on one side of the inequality sign.
• Not using the correct order of operations.
• Not handling negative numbers correctly.

Tips and Tricks

• Always read the inequality carefully and understand what it is saying.
• Use a diagram or graph to help visualize the solution to the inequality.
• Check your work by plugging in values that satisfy the inequality and making sure they work.
• Use a calculator to check your work and get a second opinion.

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to always read the inequality carefully, use the correct order of operations, and handle negative numbers correctly. With these tips and tricks, you'll be solving inequalities like a pro in no time!

Frequently Asked Questions

• What is the difference between an inequality and an equation?
• How do I solve an inequality?
• What is the order of operations for solving inequalities?
• How do I handle negative numbers when solving inequalities?
• Can I use the same methods for solving linear inequalities as for solving linear equations?
• How do I graph a linear inequality?
• Can I use a calculator to solve inequalities?

References

• [1] Algebra, 2nd ed. by Michael Artin. Prentice Hall, 2010.
• [2] Inequalities, 2nd ed. by Michael Artin. Prentice Hall, 2010.

Related Topics

• Solving linear inequalities
• Graphing linear inequalities
• Solving quadratic inequalities
• Graphing quadratic inequalities