Solve Each Compound Inequality. Then Graph The Solution Set.28. $w + 3 \leq 0$ Or $w + 7 \geq 9$30. $p - 2 \leq -2$ Or $p - 2 \ \textgreater \ 1$

Introduction

In mathematics, inequalities are used to compare two or more values. Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." In this article, we will focus on solving compound inequalities and graphing the solution set. We will use two examples to illustrate the process: $w + 3 \leq 0$ or $w + 7 \geq 9$ and $p - 2 \leq -2$ or $p - 2 > 1$.

Understanding Compound Inequalities

Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." When the word "and" is used, both inequalities must be true. When the word "or" is used, at least one of the inequalities must be true. In the first example, $w + 3 \leq 0$ or $w + 7 \geq 9$, we have two inequalities joined by the word "or." This means that either $w + 3 \leq 0$ or $w + 7 \geq 9$ must be true.

Solving the First Compound Inequality

To solve the compound inequality $w + 3 \leq 0$ or $w + 7 \geq 9$, we need to solve each inequality separately.

Solving the First Inequality

The first inequality is $w + 3 \leq 0$. To solve for $w$, we need to isolate $w$ on one side of the inequality. We can do this by subtracting 3 from both sides of the inequality.

$$w + 3 \leq 0$$

Subtracting 3 from both sides gives us:

$$w \leq -3$$

This means that $w$ is less than or equal to -3.

Solving the Second Inequality

The second inequality is $w + 7 \geq 9$. To solve for $w$, we need to isolate $w$ on one side of the inequality. We can do this by subtracting 7 from both sides of the inequality.

$$w + 7 \geq 9$$

Subtracting 7 from both sides gives us:

$$w \geq 2$$

This means that $w$ is greater than or equal to 2.

Graphing the Solution Set

To graph the solution set, we need to graph the two inequalities on a number line. The first inequality, $w \leq -3$, is a closed circle on the number line at -3, and the shading extends to the left of -3. The second inequality, $w \geq 2$, is an open circle on the number line at 2, and the shading extends to the right of 2.

Solving the Second Compound Inequality

To solve the compound inequality $p - 2 \leq -2$ or $p - 2 > 1$, we need to solve each inequality separately.

Solving the First Inequality

The first inequality is $p - 2 \leq -2$. To solve for $p$, we need to isolate $p$ on one side of the inequality. We can do this by adding 2 to both sides of the inequality.

$$p - 2 \leq -2$$

Adding 2 to both sides gives us:

$$p \leq 0$$

This means that $p$ is less than or equal to 0.

Solving the Second Inequality

The second inequality is $p - 2 > 1$. To solve for $p$, we need to isolate $p$ on one side of the inequality. We can do this by adding 2 to both sides of the inequality.

$$p - 2 > 1$$

Adding 2 to both sides gives us:

$$p > 3$$

This means that $p$ is greater than 3.

Graphing the Solution Set

To graph the solution set, we need to graph the two inequalities on a number line. The first inequality, $p \leq 0$, is a closed circle on the number line at 0, and the shading extends to the left of 0. The second inequality, $p > 3$, is an open circle on the number line at 3, and the shading extends to the right of 3.

Conclusion

In this article, we have learned how to solve compound inequalities and graph the solution set. We have used two examples to illustrate the process: $w + 3 \leq 0$ or $w + 7 \geq 9$ and $p - 2 \leq -2$ or $p - 2 > 1$. We have learned how to solve each inequality separately and graph the solution set on a number line. By following these steps, we can solve compound inequalities and graph the solution set.

Tips and Tricks

• When solving compound inequalities, make sure to solve each inequality separately.
• When graphing the solution set, make sure to graph the two inequalities on a number line.
• When graphing the solution set, make sure to shade the correct region on the number line.

Frequently Asked Questions

• Q: What is a compound inequality? A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or."
• Q: How do I solve a compound inequality? A: To solve a compound inequality, you need to solve each inequality separately.
• Q: How do I graph the solution set? A: To graph the solution set, you need to graph the two inequalities on a number line and shade the correct region.

References

• [1] Algebra and Trigonometry by Michael Sullivan
• [2] College Algebra by James Stewart
• [3] Calculus by Michael Spivak

Further Reading

• [1] Solving Linear Inequalities
• [2] Graphing Linear Inequalities
• [3] Solving Quadratic Inequalities
  

Introduction

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

Q&A

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or."

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately. If the compound inequality is joined by the word "and," both inequalities must be true. If the compound inequality is joined by the word "or," at least one of the inequalities must be true.

Q: How do I graph the solution set of a compound inequality?

A: To graph the solution set of a compound inequality, you need to graph the two inequalities on a number line and shade the correct region. If the compound inequality is joined by the word "and," you need to shade the region where both inequalities overlap. If the compound inequality is joined by the word "or," you need to shade the region where at least one of the inequalities is true.

Q: What is the difference between a compound inequality and a system of inequalities?

A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or." A system of inequalities is a set of two or more inequalities that must be true simultaneously.

Q: How do I determine whether a compound inequality is joined by the word "and" or "or"?

A: To determine whether a compound inequality is joined by the word "and" or "or," you need to look at the words used to join the inequalities. If the word "and" is used, both inequalities must be true. If the word "or" is used, at least one of the inequalities must be true.

Q: Can I use the same method to solve a compound inequality as I would to solve a system of inequalities?

A: No, you cannot use the same method to solve a compound inequality as you would to solve a system of inequalities. A compound inequality is a combination of two or more inequalities joined by the words "and" or "or," while a system of inequalities is a set of two or more inequalities that must be true simultaneously.

Q: How do I know whether to use the word "and" or "or" when writing a compound inequality?

A: To determine whether to use the word "and" or "or" when writing a compound inequality, you need to look at the context of the problem. If the problem states that both inequalities must be true, you should use the word "and." If the problem states that at least one of the inequalities must be true, you should use the word "or."

Q: Can I use a compound inequality to represent a system of inequalities?

A: No, you cannot use a compound inequality to represent a system of inequalities. A compound inequality is a combination of two or more inequalities joined by the words "and" or "or," while a system of inequalities is a set of two or more inequalities that must be true simultaneously.

Q: How do I graph a compound inequality on a number line?

A: To graph a compound inequality on a number line, you need to graph the two inequalities on a number line and shade the correct region. If the compound inequality is joined by the word "and," you need to shade the region where both inequalities overlap. If the compound inequality is joined by the word "or," you need to shade the region where at least one of the inequalities is true.

Q: Can I use a compound inequality to represent a linear inequality?

A: Yes, you can use a compound inequality to represent a linear inequality. A linear inequality is an inequality that can be written in the form $ax + b \leq c$ or $ax + b \geq c$, where $a$, $b$, and $c$ are constants.

Q: How do I determine whether a compound inequality is linear or nonlinear?

A: To determine whether a compound inequality is linear or nonlinear, you need to look at the inequalities that make up the compound inequality. If the inequalities are linear, the compound inequality is linear. If the inequalities are nonlinear, the compound inequality is nonlinear.

Conclusion

In this article, we have answered some frequently asked questions about compound inequalities. We have discussed how to solve compound inequalities, how to graph the solution set, and how to determine whether a compound inequality is linear or nonlinear. By following these steps, you can solve compound inequalities and graph the solution set.

Tips and Tricks

• When solving a compound inequality, make sure to solve each inequality separately.
• When graphing the solution set, make sure to graph the two inequalities on a number line and shade the correct region.
• When determining whether a compound inequality is linear or nonlinear, make sure to look at the inequalities that make up the compound inequality.

Frequently Asked Questions

• Q: What is a compound inequality? A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or."
• Q: How do I solve a compound inequality? A: To solve a compound inequality, you need to solve each inequality separately.
• Q: How do I graph the solution set of a compound inequality? A: To graph the solution set of a compound inequality, you need to graph the two inequalities on a number line and shade the correct region.

References

• [1] Algebra and Trigonometry by Michael Sullivan
• [2] College Algebra by James Stewart
• [3] Calculus by Michael Spivak

Further Reading

• [1] Solving Linear Inequalities
• [2] Graphing Linear Inequalities
• [3] Solving Quadratic Inequalities