-5 < 3x + 4 < 10, X ∈ Z
Introduction
In this article, we will delve into the world of inequalities and explore the solution to the given inequality: -5 < 3x + 4 < 10, where x is an integer. Inequalities are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in various mathematical disciplines. In this discussion, we will break down the inequality step by step, and provide a clear and concise solution.
Understanding the Inequality
The given inequality is a compound inequality, which means it consists of two separate inequalities joined by the word "and." The first inequality is -5 < 3x + 4, and the second inequality is 3x + 4 < 10. To solve this compound inequality, we need to solve each individual inequality separately and then find the intersection of the two solutions.
Solving the First Inequality: -5 < 3x + 4
To solve the first inequality, we need to isolate the variable x. We can start by subtracting 4 from both sides of the inequality:
-5 + 4 < 3x + 4 - 4 -1 < 3x
Next, we can divide both sides of the inequality by 3 to solve for x:
(-1)/3 < (3x)/3 -1/3 < x
So, the solution to the first inequality is x > -1/3.
Solving the Second Inequality: 3x + 4 < 10
To solve the second inequality, we can start by subtracting 4 from both sides of the inequality:
3x + 4 - 4 < 10 - 4 3x < 6
Next, we can divide both sides of the inequality by 3 to solve for x:
(3x)/3 < 6/3 x < 2
So, the solution to the second inequality is x < 2.
Finding the Intersection of the Two Solutions
Now that we have solved both individual inequalities, we need to find the intersection of the two solutions. The intersection of the two solutions is the set of values that satisfy both inequalities. In this case, the intersection of the two solutions is the set of values that are greater than -1/3 and less than 2.
Writing the Final Solution
The final solution to the inequality -5 < 3x + 4 < 10, x ∈ Z is:
-1/3 < x < 2
Understanding the Solution
The solution to the inequality is a range of values that x can take. The lower bound of the range is -1/3, and the upper bound is 2. Since x is an integer, the possible values of x are the integers between -1/3 and 2, exclusive.
Finding the Possible Values of x
To find the possible values of x, we can list the integers between -1/3 and 2, exclusive. The possible values of x are:
-1, 0, 1
Conclusion
In this article, we solved the inequality -5 < 3x + 4 < 10, x ∈ Z. We broke down the inequality into two separate inequalities and solved each one separately. We then found the intersection of the two solutions and wrote the final solution. The final solution is a range of values that x can take, and we found the possible values of x by listing the integers between -1/3 and 2, exclusive.
Frequently Asked Questions
- Q: What is the solution to the inequality -5 < 3x + 4 < 10, x ∈ Z? A: The solution to the inequality is -1/3 < x < 2.
- Q: What are the possible values of x? A: The possible values of x are -1, 0, and 1.
- Q: Why is x an integer? A: x is an integer because the problem states that x ∈ Z, which means x is an element of the set of integers.
Final Thoughts
In this article, we solved the inequality -5 < 3x + 4 < 10, x ∈ Z. We broke down the inequality into two separate inequalities and solved each one separately. We then found the intersection of the two solutions and wrote the final solution. The final solution is a range of values that x can take, and we found the possible values of x by listing the integers between -1/3 and 2, exclusive. We hope that this article has provided a clear and concise solution to the inequality and has helped you understand the concept of inequalities.
Introduction
In our previous article, we solved the inequality -5 < 3x + 4 < 10, x ∈ Z. We broke down the inequality into two separate inequalities and solved each one separately. We then found the intersection of the two solutions and wrote the final solution. In this article, we will answer some frequently asked questions about the inequality and provide additional insights.
Q&A
Q: What is the solution to the inequality -5 < 3x + 4 < 10, x ∈ Z?
A: The solution to the inequality is -1/3 < x < 2.
Q: Why is x an integer?
A: x is an integer because the problem states that x ∈ Z, which means x is an element of the set of integers.
Q: What are the possible values of x?
A: The possible values of x are -1, 0, and 1.
Q: How do I know if an inequality is true or false?
A: To determine if an inequality is true or false, you need to plug in a value for the variable and see if the inequality holds true. For example, if you plug in x = 0 into the inequality -5 < 3x + 4 < 10, you get -5 < 3(0) + 4 < 10, which simplifies to -5 < 4 < 10. Since this is true, the inequality is true for x = 0.
Q: Can I use a calculator to solve inequalities?
A: Yes, you can use a calculator to solve inequalities. However, you need to be careful when using a calculator to solve inequalities, as it may not always give you the correct solution. For example, if you use a calculator to solve the inequality -5 < 3x + 4 < 10, it may give you a decimal solution, but the solution may not be an integer.
Q: How do I know if an inequality is a compound inequality?
A: A compound inequality is an inequality that consists of two separate inequalities joined by the word "and." For example, the inequality -5 < 3x + 4 < 10 is a compound inequality, as it consists of two separate inequalities: -5 < 3x + 4 and 3x + 4 < 10.
Q: Can I use the same method to solve other inequalities?
A: Yes, you can use the same method to solve other inequalities. The method we used to solve the inequality -5 < 3x + 4 < 10 is a general method that can be used to solve other inequalities.
Q: What if the inequality has a variable on both sides?
A: If the inequality has a variable on both sides, you need to isolate the variable on one side of the inequality. For example, if you have the inequality 2x + 3 > 5, you can subtract 3 from both sides to get 2x > 2, and then divide both sides by 2 to get x > 1.
Conclusion
In this article, we answered some frequently asked questions about the inequality -5 < 3x + 4 < 10, x ∈ Z. We provided additional insights and explained how to solve other inequalities using the same method. We hope that this article has been helpful in understanding the concept of inequalities and how to solve them.
Final Thoughts
In this article, we provided a Q&A section to answer some frequently asked questions about the inequality -5 < 3x + 4 < 10, x ∈ Z. We hope that this article has been helpful in understanding the concept of inequalities and how to solve them. If you have any further questions or need additional help, please don't hesitate to ask.