Solve the Second Inequality

Oct 13, 2025 by miodeya.com 86 views

Introduction

In mathematics, inequalities are used to compare the values of two or more expressions. Compound inequalities involve multiple inequalities joined by logical operators such as "or" or "and." In this article, we will focus on solving compound inequalities, specifically the compound inequality given in the problem statement: $2x - 2 > -8$ or $-4x \leq -36$ . We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Problem

The given compound inequality is:

$2x - 2 > -8$ or $-4x \leq -36$

To solve this compound inequality, we need to solve each inequality separately and then combine the solutions using the logical operator "or."

Step 1: Solve the First Inequality

The first inequality is $2x - 2 > -8$ . To solve this inequality, we need to isolate the variable $x$ .

Isolate the Variable

First, we add 2 to both sides of the inequality:

$2x - 2 + 2 > -8 + 2$

This simplifies to:

$2x > -6$

Next, we divide both sides of the inequality by 2:

$\frac{2x}{2} > \frac{-6}{2}$

This simplifies to:

$x > -3$

Solve the Second Inequality

The second inequality is $-4x \leq -36$ . To solve this inequality, we need to isolate the variable $x$ .

Isolate the Variable

First, we divide both sides of the inequality by -4:

$\frac{-4x}{-4} \leq \frac{-36}{-4}$

This simplifies to:

$x \geq 9$

Combine the Solutions

Now that we have solved each inequality separately, we need to combine the solutions using the logical operator "or." The solution to the compound inequality is the union of the solutions to each inequality.

Union of Solutions

The solution to the first inequality is $x > -3$ , and the solution to the second inequality is $x \geq 9$ . Since the logical operator is "or," we need to find the values of $x$ that satisfy either inequality.

The values of $x$ that satisfy the first inequality are $x > -3$ , which includes all values greater than -3.

The values of $x$ that satisfy the second inequality are $x \geq 9$ , which includes all values greater than or equal to 9.

Final Solution

The final solution to the compound inequality is the union of the solutions to each inequality:

$x > -3$ or $x \geq 9$

This can be written as:

$-3 < x \leq 9$

Conclusion

In this article, we solved a compound inequality using the logical operator "or." We broke down the solution step by step and provided a clear explanation of each step. The final solution to the compound inequality is $-3 < x \leq 9$ . We hope this article has provided a clear understanding of how to solve compound inequalities.

Discussion

Do you have any questions about solving compound inequalities? Have you encountered any difficulties in solving compound inequalities? Share your thoughts and experiences in the comments below.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart
[3] "Mathematics for the Nonmathematician" by Morris Kline
Solving Compound Inequalities: Q&A =====================================

Introduction

In our previous article, we solved a compound inequality using the logical operator "or." 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 about solving compound inequalities.

Q: What is a compound inequality?

A: A compound inequality is an inequality that involves multiple inequalities joined by logical operators such as "or" or "and." For example, the compound inequality $2x - 2 > -8$ or $-4x \leq -36$ involves two inequalities joined by the logical operator "or."

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions using the logical operator. For example, to solve the compound inequality $2x - 2 > -8$ or $-4x \leq -36$ , you need to solve each inequality separately and then combine the solutions.

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

A: In compound inequalities, "or" means that the solution is the union of the solutions to each inequality. For example, the compound inequality $2x - 2 > -8$ or $-4x \leq -36$ means that the solution is the union of the solutions to each inequality. On the other hand, "and" means that the solution is the intersection of the solutions to each inequality.

Q: How do I know which logical operator to use?

A: The logical operator to use depends on the context of the problem. If the problem states "or," you should use the logical operator "or." If the problem states "and," you should use the logical operator "and."

Q: Can I use a Venn diagram to solve compound inequalities?

A: Yes, you can use a Venn diagram to solve compound inequalities. A Venn diagram is a visual representation of the solution to a compound inequality. It can help you to see the relationship between the solutions to each inequality.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you need to graph each inequality separately and then combine the graphs using the logical operator. For example, to graph the compound inequality $2x - 2 > -8$ or $-4x \leq -36$ , you need to graph each inequality separately and then combine the graphs.

Q: Can I use a calculator to solve compound inequalities?

A: Yes, you can use a calculator to solve compound inequalities. Many calculators have built-in functions for solving inequalities, including compound inequalities.

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 combining the solutions using the logical operator
Not checking the solution for consistency
Not using a Venn diagram or graph to visualize the solution

Conclusion

In this article, we answered some frequently asked questions about solving compound inequalities. We hope this article has provided a clear understanding of how to solve compound inequalities and has helped to address some common misconceptions.

Discussion

Do you have any questions about solving compound inequalities? Have you encountered any difficulties in solving compound inequalities? Share your thoughts and experiences in the comments below.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart
[3] "Mathematics for the Nonmathematician" by Morris Kline