Solve The Inequality: X 2 ≤ 1 \frac{x}{2} \leq 1 2 X ​ ≤ 1 A. X ≤ − 2 X \leq -2 X ≤ − 2 B. X ≥ 2 X \geq 2 X ≥ 2 C. X ≥ − 2 X \geq -2 X ≥ − 2 D. X ≤ 3 X \leq 3 X ≤ 3

=====================================================

Introduction

---

In mathematics, inequalities are a fundamental concept that helps us compare values and make decisions based on certain conditions. Solving an inequality involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $\frac{x}{2} \leq 1$ and explore the different solution sets.

Understanding the Inequality

---

The given inequality is $\frac{x}{2} \leq 1$. To solve this inequality, we need to isolate the variable $x$. We can start by multiplying both sides of the inequality by 2, which is a positive number. This will help us get rid of the fraction and make it easier to solve.

Multiplying Both Sides

---

\frac{x}{2} \leq 1

Multiplying both sides by 2:

x \leq 2

Analyzing the Solution

---

Now that we have the inequality in the form $x \leq 2$, we can analyze the solution. The solution set includes all values of $x$ that satisfy the inequality. In this case, the solution set is all real numbers less than or equal to 2.

Solution Set

---

The solution set is $x \leq 2$. This means that any value of $x$ that is less than or equal to 2 will satisfy the inequality.

Checking the Answer Choices

---

Now that we have the solution set, we can check the answer choices to see which one matches our solution.

Answer Choice A

---

$x \leq -2$

This answer choice does not match our solution set. The solution set includes all values of $x$ that are less than or equal to 2, not less than or equal to -2.

Answer Choice B

---

$x \geq 2$

This answer choice does not match our solution set. The solution set includes all values of $x$ that are less than or equal to 2, not greater than or equal to 2.

Answer Choice C

---

$x \geq -2$

This answer choice does not match our solution set. The solution set includes all values of $x$ that are less than or equal to 2, not greater than or equal to -2.

Answer Choice D

---

$x \leq 3$

This answer choice does not match our solution set. The solution set includes all values of $x$ that are less than or equal to 2, not less than or equal to 3.

Conclusion

---

In conclusion, the solution to the inequality $\frac{x}{2} \leq 1$ is $x \leq 2$. This means that any value of $x$ that is less than or equal to 2 will satisfy the inequality.

Frequently Asked Questions

---

Q: What is the solution to the inequality $\frac{x}{2} \leq 1$?

A: The solution to the inequality $\frac{x}{2} \leq 1$ is $x \leq 2$.

Q: What is the solution set for the inequality $\frac{x}{2} \leq 1$?

A: The solution set for the inequality $\frac{x}{2} \leq 1$ is all real numbers less than or equal to 2.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing operations on both sides of the inequality.

Final Answer

---

The final answer is $x \leq 2$.

=====================================================

Introduction

---

In our previous article, we explored the concept of solving inequalities and provided a step-by-step guide on how to solve the inequality $\frac{x}{2} \leq 1$. In this article, we will delve deeper into the world of inequalities and provide a comprehensive Q&A guide to help you understand and solve inequalities with ease.

Q&A: Solving Inequalities

---

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two values using a comparison operator such as <, >, ≤, or ≥.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities involve a linear expression, while quadratic inequalities involve a quadratic expression.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable by performing operations on both sides of the inequality. You can add or subtract the same value from both sides, or multiply or divide both sides by the same non-zero value.

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 quadratic expression to determine the solution set.

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

A: A linear inequality involves a linear expression, while a quadratic inequality involves a quadratic expression. Linear inequalities are typically easier to solve than quadratic inequalities.

Q: Can I use the same methods to solve all types of inequalities?

A: No, the methods used to solve different types of inequalities may vary. For example, you may need to use factoring to solve a quadratic inequality, while you may need to use the quadratic formula to solve a quadratic inequality.

Q&A: Inequality Operations

---

Q: Can I add or subtract the same value from both sides of an inequality?

A: Yes, you can add or subtract the same value from both sides of an inequality. This is known as adding or subtracting a constant.

Q: Can I multiply or divide both sides of an inequality by the same non-zero value?

A: Yes, you can multiply or divide both sides of an inequality by the same non-zero value. This is known as multiplying or dividing by a constant.

Q: What happens if I multiply or divide both sides of an inequality by a negative value?

A: If you multiply or divide both sides of an inequality by a negative value, you need to reverse the direction of the inequality.

Q: Can I multiply or divide both sides of an inequality by a variable?

A: No, you cannot multiply or divide both sides of an inequality by a variable. This is because the variable may be positive or negative, and you need to consider both cases.

Q&A: Inequality Solutions

---

Q: What is the solution set for an inequality?

A: The solution set for an inequality is the set of all values that satisfy the inequality.

Q: How do I determine the solution set for an inequality?

A: To determine the solution set for an inequality, you need to isolate the variable and then use the sign of the inequality to determine the solution set.

Q: Can I have multiple solution sets for an inequality?

A: Yes, you can have multiple solution sets for an inequality. This occurs when the inequality has multiple critical points.

Q: How do I graph an inequality?

A: To graph an inequality, you need to plot the critical points on a number line and then use the sign of the inequality to determine the solution set.

Conclusion

---

In conclusion, solving inequalities requires a deep understanding of the different types of inequalities, inequality operations, and inequality solutions. By following the steps outlined in this article, you can become proficient in solving inequalities and apply your knowledge to real-world problems.

Frequently Asked Questions

---

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

A: A linear inequality involves a linear expression, while a quadratic inequality involves a quadratic expression.

Q: Can I use the same methods to solve all types of inequalities?

A: No, the methods used to solve different types of inequalities may vary.

Q: How do I determine the solution set for an inequality?

A: To determine the solution set for an inequality, you need to isolate the variable and then use the sign of the inequality to determine the solution set.

Final Answer

---

The final answer is that solving inequalities requires a deep understanding of the different types of inequalities, inequality operations, and inequality solutions. By following the steps outlined in this article, you can become proficient in solving inequalities and apply your knowledge to real-world problems.