Which Compound Inequality Has No Solution?A. $x \leq -2$ And $2x \geq 6$B. $x \leq -1$ And $5x \leq 5$C. $x \leq -1$ And $3x \geq -3$D. $x \leq -2$ And $4x \leq -8$

Introduction

In mathematics, inequalities are used to describe the relationship between two or more values. Compound inequalities are a combination of two or more inequalities that are connected by logical operators such as "and" or "or". In this article, we will explore which compound inequality has no solution.

Understanding Compound Inequalities

A compound inequality is a statement that combines two or more inequalities using logical operators. For example, the compound inequality "x ≤ -2 and 2x ≥ 6" is a combination of two inequalities: x ≤ -2 and 2x ≥ 6. To solve a compound inequality, we need to find the values of x that satisfy both inequalities.

Analyzing the Options

Let's analyze each option to determine which compound inequality has no solution.

Option A: $x \leq -2$ and $2x \geq 6$

To solve this compound inequality, we need to find the values of x that satisfy both inequalities. We can start by solving the first inequality: x ≤ -2. This means that x is less than or equal to -2.

Next, we can solve the second inequality: 2x ≥ 6. To do this, we can divide both sides of the inequality by 2, which gives us x ≥ 3. This means that x is greater than or equal to 3.

Since the two inequalities are contradictory (x cannot be both less than or equal to -2 and greater than or equal to 3 at the same time), this compound inequality has no solution.

Option B: $x \leq -1$ and $5x \leq 5$

To solve this compound inequality, we need to find the values of x that satisfy both inequalities. We can start by solving the first inequality: x ≤ -1. This means that x is less than or equal to -1.

Next, we can solve the second inequality: 5x ≤ 5. To do this, we can divide both sides of the inequality by 5, which gives us x ≤ 1. This means that x is less than or equal to 1.

Since the two inequalities are not contradictory (x can be less than or equal to -1 and less than or equal to 1 at the same time), this compound inequality has a solution.

Option C: $x \leq -1$ and $3x \geq -3$

To solve this compound inequality, we need to find the values of x that satisfy both inequalities. We can start by solving the first inequality: x ≤ -1. This means that x is less than or equal to -1.

Next, we can solve the second inequality: 3x ≥ -3. To do this, we can divide both sides of the inequality by 3, which gives us x ≥ -1. This means that x is greater than or equal to -1.

Since the two inequalities are not contradictory (x can be less than or equal to -1 and greater than or equal to -1 at the same time), this compound inequality has a solution.

Option D: $x \leq -2$ and $4x \leq -8$

To solve this compound inequality, we need to find the values of x that satisfy both inequalities. We can start by solving the first inequality: x ≤ -2. This means that x is less than or equal to -2.

Next, we can solve the second inequality: 4x ≤ -8. To do this, we can divide both sides of the inequality by 4, which gives us x ≤ -2. This means that x is less than or equal to -2.

Since the two inequalities are not contradictory (x can be less than or equal to -2 and less than or equal to -2 at the same time), this compound inequality has a solution.

Conclusion

In conclusion, the compound inequality that has no solution is Option A: $x \leq -2$ and $2x \geq 6$. This is because the two inequalities are contradictory, and there is no value of x that can satisfy both inequalities at the same time.

Frequently Asked Questions

• What is a compound inequality? A compound inequality is a statement that combines two or more inequalities using logical operators.
• How do I solve a compound inequality? To solve a compound inequality, you need to find the values of x that satisfy both inequalities.
• What is the difference between a compound inequality and a single inequality? A compound inequality is a combination of two or more inequalities, while a single inequality is a statement that describes the relationship between two values.

Final Thoughts

In this article, we explored which compound inequality has no solution. We analyzed each option and determined that Option A: $x \leq -2$ and $2x \geq 6$ has no solution. This is because the two inequalities are contradictory, and there is no value of x that can satisfy both inequalities at the same time. We hope that this article has provided you with a better understanding of compound inequalities and how to solve them.

Introduction

In our previous article, we explored which compound inequality has no solution. We analyzed each option and determined that Option A: $x \leq -2$ and $2x \geq 6$ has no solution. In this article, we will provide a Q&A section to help you better understand compound inequalities and how to solve them.

Q&A

Q: What is a compound inequality?

A: A compound inequality is a statement that combines two or more inequalities using logical operators.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to find the values of x that satisfy both inequalities. You can do this by solving each inequality separately and then finding the intersection of the two solutions.

Q: What is the difference between a compound inequality and a single inequality?

A: A compound inequality is a combination of two or more inequalities, while a single inequality is a statement that describes the relationship between two values.

Q: Can a compound inequality have multiple solutions?

A: Yes, a compound inequality can have multiple solutions. For example, the compound inequality $x \leq -2$ and $x \geq 3$ has no solution, but the compound inequality $x \leq -2$ and $x \geq 0$ has multiple solutions.

Q: How do I know if a compound inequality has a solution or not?

A: To determine if a compound inequality has a solution or not, you need to find the intersection of the two solutions. If the intersection is empty, then the compound inequality has no solution.

Q: Can a compound inequality be true for all values of x?

A: No, a compound inequality cannot be true for all values of x. If a compound inequality is true for all values of x, then it is not a compound inequality, but rather a single inequality.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you need to graph each inequality separately and then find the intersection of the two graphs.

Q: Can a compound inequality be used to describe a set of values?

A: Yes, a compound inequality can be used to describe a set of values. For example, the compound inequality $x \leq -2$ and $x \geq 3$ describes the set of values that are less than or equal to -2 and greater than or equal to 3.

Q: How do I write a compound inequality in interval notation?

A: To write a compound inequality in interval notation, you need to find the intersection of the two solutions and then write it in interval notation.

Conclusion

In this article, we provided a Q&A section to help you better understand compound inequalities and how to solve them. We hope that this article has provided you with a better understanding of compound inequalities and how to use them to describe sets of values.

Frequently Asked Questions

• What is a compound inequality?
• How do I solve a compound inequality?
• What is the difference between a compound inequality and a single inequality?
• Can a compound inequality have multiple solutions?
• How do I know if a compound inequality has a solution or not?
• Can a compound inequality be true for all values of x?
• How do I graph a compound inequality?
• Can a compound inequality be used to describe a set of values?
• How do I write a compound inequality in interval notation?

Final Thoughts

In this article, we provided a Q&A section to help you better understand compound inequalities and how to solve them. We hope that this article has provided you with a better understanding of compound inequalities and how to use them to describe sets of values.