Explain Why The Solution Of The Compound Inequality $5x - 3 \ \textgreater \ 14.5$ Or $\frac{2x + 5}{3} \ \textless \ 4$ Includes All Real Numbers, With One Exception.

Introduction

Compound inequalities involve multiple inequalities combined using logical operators such as "or" or "and." Solving compound inequalities requires careful consideration of each individual inequality and how they interact with each other. In this article, we will explore the solution to the compound inequality $5x - 3 \ \textgreater \ 14.5$ or $\frac{2x + 5}{3} \ \textless \ 4$ and explain why it includes all real numbers, with one exception.

Solving the Compound Inequality

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

Solving the First Inequality

The first inequality is $5x - 3 \ \textgreater \ 14.5$. To solve for $x$, we can add 3 to both sides of the inequality:

$$5x - 3 + 3 \ \textgreater \ 14.5 + 3$$

This simplifies to:

$$5x \ \textgreater \ 17.5$$

Next, we can divide both sides of the inequality by 5:

$$\frac{5x}{5} \ \textgreater \ \frac{17.5}{5}$$

This simplifies to:

$$x \ \textgreater \ 3.5$$

Solving the Second Inequality

The second inequality is $\frac{2x + 5}{3} \ \textless \ 4$. To solve for $x$, we can multiply both sides of the inequality by 3:

$$\frac{2x + 5}{3} \cdot 3 \ \textless \ 4 \cdot 3$$

This simplifies to:

$$2x + 5 \ \textless \ 12$$

Next, we can subtract 5 from both sides of the inequality:

$$2x + 5 - 5 \ \textless \ 12 - 5$$

This simplifies to:

$$2x \ \textless \ 7$$

Finally, we can divide both sides of the inequality by 2:

$$\frac{2x}{2} \ \textless \ \frac{7}{2}$$

This simplifies to:

$$x \ \textless \ 3.5$$

Combining the Solutions

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

$$x \ \textgreater \ 3.5 \ \text{or} \ x \ \textless \ 3.5$$

This can be rewritten as:

$$x \ \textgreater \ 3.5 \ \text{or} \ x \ \textless \ 3.5$$

Understanding the Exception

At first glance, it may seem that the solution to the compound inequality includes all real numbers. However, upon closer inspection, we can see that the solution is actually all real numbers except for one value: $x = 3.5$.

To understand why this is the case, let's consider what happens when we substitute $x = 3.5$ into each individual inequality. For the first inequality, we have:

$$5(3.5) - 3 \ \textgreater \ 14.5$$

This simplifies to:

$$17.5 - 3 \ \textgreater \ 14.5$$

This is a true statement, so $x = 3.5$ satisfies the first inequality.

For the second inequality, we have:

$$\frac{2(3.5) + 5}{3} \ \textless \ 4$$

This simplifies to:

$$\frac{7 + 5}{3} \ \textless \ 4$$

This simplifies to:

$$\frac{12}{3} \ \textless \ 4$$

This is a false statement, so $x = 3.5$ does not satisfy the second inequality.

Since $x = 3.5$ satisfies the first inequality but not the second, it is not included in the solution to the compound inequality.

Conclusion

$$$$$$$$

Introduction

In our previous article, we explored the solution to the compound inequality $5x - 3 \ \textgreater \ 14.5$ or $\frac{2x + 5}{3} \ \textless \ 4$ and explained why it includes all real numbers, with one exception. In this article, we will answer some frequently asked questions about compound inequalities and provide additional examples to help solidify your understanding.

Q&A

Q: What is a compound inequality?

A: A compound inequality is an inequality that involves multiple inequalities combined using logical operators such as "or" or "and." For example, the compound inequality $5x - 3 \ \textgreater \ 14.5$ or $\frac{2x + 5}{3} \ \textless \ 4$ involves two individual inequalities combined using the logical operator "or."

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each individual inequality separately and then combine the solutions using the logical operator. For example, to solve the compound inequality $5x - 3 \ \textgreater \ 14.5$ or $\frac{2x + 5}{3} \ \textless \ 4$, you would first solve the individual inequalities $5x - 3 \ \textgreater \ 14.5$ and $\frac{2x + 5}{3} \ \textless \ 4$ separately, and then combine the solutions using the logical operator "or."

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

A: A compound inequality involves multiple inequalities combined using logical operators, while a system of inequalities involves multiple inequalities that must all be satisfied simultaneously. For example, the compound inequality $5x - 3 \ \textgreater \ 14.5$ or $\frac{2x + 5}{3} \ \textless \ 4$ involves two individual inequalities combined using the logical operator "or," while the system of inequalities $x \ \textgreater \ 2$ and $x \ \textless \ 5$ involves two individual inequalities that must both be satisfied simultaneously.

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

A: No, the method for solving a compound inequality is different from the method for solving a system of inequalities. To solve a compound inequality, you need to solve each individual inequality separately and then combine the solutions using the logical operator. To solve a system of inequalities, you need to find the intersection of the solutions to each individual inequality.

Q: What is the exception in the solution to the compound inequality $5x - 3 \ \textgreater \ 14.5$ or $\frac{2x + 5}{3} \ \textless \ 4$?

A: The exception in the solution to the compound inequality $5x - 3 \ \textgreater \ 14.5$ or $\frac{2x + 5}{3} \ \textless \ 4$ is the value $x = 3.5$. This is because $x = 3.5$ satisfies the first inequality but not the second, and the solution to the compound inequality is the union of the solutions to each individual inequality.

Q: Can I use the same method to solve a compound inequality with more than two individual inequalities?

A: Yes, you can use the same method to solve a compound inequality with more than two individual inequalities. For example, to solve the compound inequality $5x - 3 \ \textgreater \ 14.5$ or $\frac{2x + 5}{3} \ \textless \ 4$ or $x \ \textless \ 2$, you would first solve the individual inequalities $5x - 3 \ \textgreater \ 14.5$, $\frac{2x + 5}{3} \ \textless \ 4$, and $x \ \textless \ 2$ separately, and then combine the solutions using the logical operator "or."

Additional Examples

Example 1

Solve the compound inequality $2x + 5 \ \textless \ 10$ or $x - 3 \ \textgreater \ 2$.

Solution

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

First, we solve the individual inequality $2x + 5 \ \textless \ 10$:

$$2x + 5 - 5 \ \textless \ 10 - 5$$

This simplifies to:

$$2x \ \textless \ 5$$

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

$$\frac{2x}{2} \ \textless \ \frac{5}{2}$$

This simplifies to:

$$x \ \textless \ 2.5$$

Next, we solve the individual inequality $x - 3 \ \textgreater \ 2$:

$$x - 3 + 3 \ \textgreater \ 2 + 3$$

This simplifies to:

$$x \ \textgreater \ 5$$

Finally, we combine the solutions using the logical operator "or":

$$x \ \textless \ 2.5 \ \text{or} \ x \ \textgreater \ 5$$

Example 2

Solve the compound inequality $x + 2 \ \textless \ 5$ or $x - 2 \ \textgreater \ 3$.

Solution

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

First, we solve the individual inequality $x + 2 \ \textless \ 5$:

$$x + 2 - 2 \ \textless \ 5 - 2$$

This simplifies to:

$$x \ \textless \ 3$$

Next, we solve the individual inequality $x - 2 \ \textgreater \ 3$:

$$x - 2 + 2 \ \textgreater \ 3 + 2$$

This simplifies to:

$$x \ \textgreater \ 5$$

Finally, we combine the solutions using the logical operator "or":

$$x \ \textless \ 3 \ \text{or} \ x \ \textgreater \ 5$$

Conclusion

In conclusion, compound inequalities involve multiple inequalities combined using logical operators, and solving them requires careful consideration of each individual inequality and how they interact with each other. By following the steps outlined in this article, you can solve compound inequalities and understand the exceptions in their solutions.