What Is The Solution Set To The Compound Inequality 2 X − 7 \textgreater 3 2x - 7 \ \textgreater \ 3 2 X − 7 \textgreater 3 Or 4 − X ≤ − 4 4 - X \leq -4 4 − X ≤ − 4 ?A. All Real Numbers B. X \textgreater 5 X \ \textgreater \ 5 X \textgreater 5 C. 5 \textless X ≤ 8 5 \ \textless \ X \leq 8 5 \textless X ≤ 8 D. X ≤ 8 X \leq 8 X ≤ 8

Introduction

Compound inequalities are a combination of two or more inequalities joined by the words "or" or "and." In this article, we will focus on solving the compound inequality $2x - 7 \ \textgreater \ 3$ or $4 - x \leq -4$. To solve this compound inequality, we need to solve each inequality separately and then find the solution set that satisfies both inequalities.

Step 1: Solve the First Inequality $2x - 7 \ \textgreater \ 3$

To solve the first inequality, we need to isolate the variable x. We can do this by adding 7 to both sides of the inequality.

2x - 7 + 7 > 3 + 7
2x > 10

Next, we need to divide both sides of the inequality by 2 to solve for x.

(2x) / 2 > 10 / 2
x > 5

Step 2: Solve the Second Inequality $4 - x \leq -4$

To solve the second inequality, we need to isolate the variable x. We can do this by subtracting 4 from both sides of the inequality.

4 - x - 4 <= -4 - 4
-x <= -8

Next, we need to multiply both sides of the inequality by -1 to solve for x. Remember that when we multiply or divide an inequality by a negative number, we need to reverse the direction of the inequality.

(-x) * (-1) >= (-8) * (-1)
x >= 8

Step 3: Find the Solution Set that Satisfies Both Inequalities

Now that we have solved both inequalities, we need to find the solution set that satisfies both inequalities. We can do this by finding the intersection of the two solution sets.

x > 5 and x >= 8

However, we need to find the intersection of the two solution sets, which means we need to find the values of x that satisfy both inequalities. In this case, the solution set is the values of x that are greater than 5 and also greater than or equal to 8.

Conclusion

The solution set to the compound inequality $2x - 7 \ \textgreater \ 3$ or $4 - x \leq -4$ is the values of x that are greater than 5 and also greater than or equal to 8. This can be written as $5 \ \textless \ x \leq 8$.

Final Answer

The final answer is $5 \ \textless \ x \leq 8$.

Introduction

Compound inequalities are a combination of two or more inequalities joined by the words "or" or "and." In this article, we will answer some frequently asked questions about compound inequalities.

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities joined by the words "or" or "and." For example, $2x - 7 \ \textgreater \ 3$ or $4 - x \leq -4$ is a compound inequality.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then find the solution set that satisfies both inequalities. You can use the steps outlined in the previous article to solve each inequality.

Q: What is the difference between "or" and "and" in compound inequalities?

A: In compound inequalities, "or" means that either of the inequalities can be true, while "and" means that both inequalities must be true. For example, $2x - 7 \ \textgreater \ 3$ or $4 - x \leq -4$ means that either $2x - 7 \ \textgreater \ 3$ or $4 - x \leq -4$ can be true, while $2x - 7 \ \textgreater \ 3$ and $4 - x \leq -4$ means that both $2x - 7 \ \textgreater \ 3$ and $4 - x \leq -4$ must be true.

Q: How do I find the solution set that satisfies both inequalities?

A: To find the solution set that satisfies both inequalities, you need to find the intersection of the two solution sets. This means that you need to find the values of x that satisfy both inequalities.

Q: What is the intersection of two solution sets?

A: The intersection of two solution sets is the set of values that are common to both solution sets. For example, if one solution set is $x > 5$ and the other solution set is $x \geq 8$, the intersection of the two solution sets is $x \geq 8$.

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 solution set that satisfies both inequalities. You can use a number line to graph each inequality and then find the intersection of the two solution sets.

Q: What are some common mistakes to avoid when solving compound inequalities?

A: Some common mistakes to avoid when solving compound inequalities include:

• Not solving each inequality separately
• Not finding the intersection of the two solution sets
• Not considering the direction of the inequality
• Not using a number line to graph the inequalities

Conclusion

Compound inequalities are a combination of two or more inequalities joined by the words "or" or "and." To solve a compound inequality, you need to solve each inequality separately and then find the solution set that satisfies both inequalities. By following the steps outlined in this article, you can solve compound inequalities and find the solution set that satisfies both inequalities.

Final Answer

The final answer is that compound inequalities are a combination of two or more inequalities joined by the words "or" or "and," and can be solved by solving each inequality separately and finding the solution set that satisfies both inequalities.