Consider The Following Compound Inequality:$0 \ \textless \ -5x + 5 \text{ Or } 3x \geq 6$1. Solve The Inequality For $x$.- Answers Of The Form $3 \ \textless \ X$ And $x \ \textless \ 5$ Should Be Entered

Introduction

In mathematics, inequalities are used to describe relationships between variables. Compound inequalities involve multiple inequalities joined by logical operators, such as "or" or "and." In this article, we will focus on solving compound inequalities of the form $a \lt x$ and $x \lt b$ or $c \geq x$ and $x \geq d$. We will use the given compound inequality $0 \lt -5x + 5$ or $3x \geq 6$ as an example to demonstrate the steps involved in solving compound inequalities.

Understanding the Compound Inequality

The given compound inequality is $0 \lt -5x + 5$ or $3x \geq 6$. This means that either the inequality $0 \lt -5x + 5$ is true, or the inequality $3x \geq 6$ is true, or both.

Breaking Down the Inequalities

To solve the compound inequality, we need to break down each inequality separately.

Inequality 1: $0 \lt -5x + 5$

To solve this inequality, we need to isolate the variable $x$. We can start by subtracting 5 from both sides of the inequality:

$$0 - 5 \lt -5x + 5 - 5$$

This simplifies to:

$$-5 \lt -5x$$

Next, we can divide both sides of the inequality by -5. However, when we divide by a negative number, the inequality sign is reversed:

$$\frac{-5}{-5} \gt \frac{-5x}{-5}$$

This simplifies to:

$$1 \gt x$$

So, the solution to the first inequality is $x \lt 1$.

Inequality 2: $3x \geq 6$

To solve this inequality, we need to isolate the variable $x$. We can start by dividing both sides of the inequality by 3:

$$\frac{3x}{3} \geq \frac{6}{3}$$

This simplifies to:

$$x \geq 2$$

So, the solution to the second inequality is $x \geq 2$.

Combining the Solutions

Now that we have solved each inequality separately, we need to combine the solutions. Since the compound inequality is "or," we need to find the union of the two solutions.

The solution to the first inequality is $x \lt 1$, and the solution to the second inequality is $x \geq 2$. To find the union of these two solutions, we need to find the values of $x$ that satisfy either of the two inequalities.

Since $x \lt 1$ and $x \geq 2$ are mutually exclusive, there is no value of $x$ that satisfies both inequalities. Therefore, the solution to the compound inequality is the union of the two solutions:

$$x \lt 1 \text{ or } x \geq 2$$

This can be written as:

$$x \in (-\infty, 1) \cup [2, \infty)$$

Conclusion

---

Solving compound inequalities involves breaking down each inequality separately and then combining the solutions. In this article, we used the compound inequality $0 \lt -5x + 5$ or $3x \geq 6$ as an example to demonstrate the steps involved in solving compound inequalities. We broke down each inequality separately, isolated the variable $x$, and then combined the solutions to find the union of the two solutions.

Tips and Tricks

• When solving compound inequalities, it's essential to break down each inequality separately and then combine the solutions.
• When combining the solutions, use the "or" operator to find the union of the two solutions.
• When solving compound inequalities, it's essential to check for mutually exclusive solutions.

Practice Problems

1. Solve the compound inequality $2x \lt 5$ or $x \geq 3$.
2. Solve the compound inequality $x \lt 2$ and $x \geq 4$.
3. Solve the compound inequality $3x \geq 6$ or $x \lt 1$.

Answer Key

1. $x \in (-\infty, \frac{5}{2}) \cup [3, \infty)$
2. There is no solution to this compound inequality.
3. $x \in (-\infty, 1) \cup [2, \infty)$
   Solving Compound Inequalities: A Q&A Guide =====================================================

Introduction

In our previous article, we discussed how to solve compound inequalities of the form $a \lt x$ and $x \lt b$ or $c \geq x$ and $x \geq d$. In this article, we will provide a Q&A guide to help you better understand how to solve 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 $0 \lt -5x + 5$ or $3x \geq 6$ involves two inequalities joined by the "or" operator.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to break down each inequality separately and then combine the solutions. Here are the steps:

1. Break down each inequality separately.
2. Isolate the variable $x$ in each inequality.
3. Combine the solutions using the "or" operator.

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

A: In compound inequalities, "or" means that either of the two inequalities can be true, while "and" means that both inequalities must be true. For example, the compound inequality $x \lt 2$ or $x \geq 3$ means that either $x$ is less than 2 or $x$ is greater than or equal to 3, while the compound inequality $x \lt 2$ and $x \geq 3$ means that $x$ is both less than 2 and greater than or equal to 3.

Q: How do I handle mutually exclusive solutions in compound inequalities?

A: When solving compound inequalities, you may encounter mutually exclusive solutions, which means that the two inequalities cannot be true at the same time. In this case, the solution to the compound inequality is the union of the two solutions.

Q: Can you provide an example of a compound inequality with mutually exclusive solutions?

A: Yes, consider the compound inequality $x \lt 1$ and $x \geq 2$. These two inequalities are mutually exclusive, which means that there is no value of $x$ that satisfies both inequalities. Therefore, the solution to the compound inequality is the union of the two solutions:

$$x \in (-\infty, 1) \cup [2, \infty)$$

Q: How do I write the solution to a compound inequality?

A: When writing the solution to a compound inequality, you need to use interval notation. For example, the solution to the compound inequality $x \lt 1$ or $x \geq 2$ can be written as:

$$x \in (-\infty, 1) \cup [2, \infty)$$

Q: Can you provide some practice problems to help me better understand how to solve compound inequalities?

A: Yes, here are some practice problems:

1. Solve the compound inequality $2x \lt 5$ or $x \geq 3$.
2. Solve the compound inequality $x \lt 2$ and $x \geq 4$.
3. Solve the compound inequality $3x \geq 6$ or $x \lt 1$.

Answer Key

1. $x \in (-\infty, \frac{5}{2}) \cup [3, \infty)$
2. There is no solution to this compound inequality.
3. $x \in (-\infty, 1) \cup [2, \infty)$

Conclusion

Solving compound inequalities involves breaking down each inequality separately and then combining the solutions. By following the steps outlined in this article, you can better understand how to solve compound inequalities and write the solution in interval notation. Remember to handle mutually exclusive solutions by finding the union of the two solutions.