Solve the first inequality

Oct 13, 2025 by windailys.com 51 views

Introduction

In mathematics, inequalities are used to describe the relationship between two or more values. A compound inequality is a combination of two or more inequalities, connected by logical operators such as "and" or "or". In this article, we will focus on identifying the graph of the compound inequality $(-2 \ \textless \ 2x - 6) \cap (3x - 12 \leq 10)$ .

Understanding the Inequality

To begin, let's break down the compound inequality into its individual components. The first inequality is $-2 \ \textless \ 2x - 6$ , which can be rewritten as $2x - 6 > -2$ . The second inequality is $3x - 12 \leq 10$ , which can be rewritten as $3x \leq 22$ .

Solving the First Inequality

To solve the first inequality, we need to isolate the variable $x$ . We can do this by adding 6 to both sides of the inequality, which gives us $2x > 4$ . Next, we can divide both sides of the inequality by 2, which gives us $x > 2$ .

Solving the Second Inequality

To solve the second inequality, we need to isolate the variable $x$ . We can do this by adding 12 to both sides of the inequality, which gives us $3x \leq 22$ . Next, we can divide both sides of the inequality by 3, which gives us $x \leq \frac{22}{3}$ .

Finding the Intersection

Now that we have solved both inequalities, we need to find the intersection of the two solution sets. The first inequality has a solution set of $x > 2$ , while the second inequality has a solution set of $x \leq \frac{22}{3}$ . To find the intersection, we need to find the values of $x$ that satisfy both inequalities.

Graphing the Solution Set

To graph the solution set, we can use a number line. We can start by marking the point $x = 2$ on the number line, which represents the boundary of the first inequality. We can then mark the point $x = \frac{22}{3}$ on the number line, which represents the boundary of the second inequality.

Conclusion

In conclusion, the compound inequality $(-2 \ \textless \ 2x - 6) \cap (3x - 12 \leq 10)$ can be solved by finding the intersection of the two solution sets. The solution set is the set of all values of $x$ that satisfy both inequalities. By graphing the solution set on a number line, we can visualize the range of values that satisfy the compound inequality.

Step-by-Step Solution

Here is a step-by-step solution to the compound inequality:

Solve the first inequality: $2x - 6 > -2$ $2 x - 6 > - 2$
- Add 6 to both sides: $2x > 4$
- Divide both sides by 2: $x > 2$
Solve the second inequality: $3x - 12 \leq 10$ $3 x - 12 \leq 10$
- Add 12 to both sides: $3x \leq 22$
- Divide both sides by 3: $x \leq \frac{22}{3}$
Find the intersection: The solution set is the set of all values of $x$ that satisfy both inequalities.
Graph the solution set: Use a number line to mark the points $x = 2$ and $x = \frac{22}{3}$ , and shade the region between them.

Example Problems

Here are some example problems that involve solving compound inequalities:

Solve the compound inequality $(-3 \ \textless \ 2x - 4) \cap (x - 2 \leq 5)$ .
Solve the compound inequality $(x + 2 > 3) \cap (2x - 1 \leq 7)$ .
Solve the compound inequality $(-1 \ \textless \ x - 3) \cap (x + 1 \leq 4)$ .

Tips and Tricks

Here are some tips and tricks for solving compound inequalities:

Use a number line: A number line can be a useful tool for visualizing the solution set of a compound inequality.
Mark the boundaries: Make sure to mark the boundaries of the solution set on the number line.
Shade the region: Shade the region between the boundaries to represent the solution set.
Check your work: Make sure to check your work by plugging in test values into the inequality.

Conclusion

In conclusion, solving compound inequalities requires a combination of algebraic skills and visual thinking. By following the steps outlined in this article, you can learn how to solve compound inequalities and visualize the solution set using a number line. With practice and patience, you can become proficient in solving compound inequalities and apply this skill to a wide range of mathematical problems.

Introduction

Compound inequalities are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. In this article, we will answer some of the most frequently asked questions about compound inequalities, providing you with a deeper understanding of this important topic.

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities, connected by logical operators such as "and" or "or". For example, the compound inequality $(-2 \ \textless \ 2x - 6) \cap (3x - 12 \leq 10)$ is a combination of two inequalities: $2x - 6 > -2$ and $3x - 12 \leq 10$ .

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each individual inequality separately and then 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 difference between a compound inequality and a single inequality?

A: A single inequality is a statement that compares two values, such as $x > 2$ . A compound inequality, on the other hand, is a combination of two or more inequalities, connected by logical operators.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you need to use a number line to mark the boundaries of the solution set. You can then shade the region between the boundaries to represent the solution set.

Q: What is the intersection of two solution sets?

A: The intersection of two solution sets is the set of all values that satisfy both inequalities. For example, if the first inequality has a solution set of $x > 2$ and the second inequality has a solution set of $x \leq 4$ , then the intersection of the two solution sets is the set of all values that satisfy both inequalities, which is $2 < x \leq 4$ .

Q: How do I find the intersection of two solution sets?

A: To find the intersection of two solution sets, you need to find the values of $x$ that satisfy both inequalities. You can do this by solving each individual inequality separately and then finding the values of $x$ that satisfy both inequalities.

Q: What is the union of two solution sets?

A: The union of two solution sets is the set of all values that satisfy either inequality. For example, if the first inequality has a solution set of $x > 2$ and the second inequality has a solution set of $x \leq 4$ , then the union of the two solution sets is the set of all values that satisfy either inequality, which is $x > 2$ or $x \leq 4$ .

Q: How do I find the union of two solution sets?

A: To find the union of two solution sets, you need to find the values of $x$ that satisfy either inequality. You can do this by solving each individual inequality separately and then finding the values of $x$ that satisfy either inequality.

Q: What is the difference between the intersection and the union of two solution sets?

A: The intersection of two solution sets is the set of all values that satisfy both inequalities, while the union of two solution sets is the set of all values that satisfy either inequality.

Q: How do I apply compound inequalities in real-life situations?

A: Compound inequalities are used in a wide range of real-life situations, such as finance, engineering, and science. For example, a company may use compound inequalities to determine the range of values for a product's price, while an engineer may use compound inequalities to determine the range of values for a system's parameters.

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

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

Failing to solve each individual inequality separately
Failing to find the intersection of the two solution sets
Failing to graph the solution set correctly
Failing to check the work for errors

Q: How can I practice solving compound inequalities?

A: You can practice solving compound inequalities by working through example problems, such as those found in this article. You can also try creating your own example problems and solving them on your own.

Q: What are some resources for learning more about compound inequalities?

A: Some resources for learning more about compound inequalities include:

Textbooks on algebra and mathematics
Online tutorials and videos
Practice problems and worksheets
Online communities and forums

Conclusion

In conclusion, compound inequalities are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. By following the steps outlined in this article and practicing regularly, you can become proficient in solving compound inequalities and apply this skill to a wide range of mathematical problems.