Select The Correct Answer.Solve The Following Inequality:${ -8 \ \textless \ 3x + 1 \ \textless \ 7 }$A. { -3 \ \textless \ X \ \textless \ 2$}$B. { -3 \ \textless \ X \ \textless \ \frac{7}{3}$}$C.

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Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. A linear inequality is an inequality that can be written in the form of ax + b < c, where a, b, and c are constants, and x is the variable. In this article, we will focus on solving linear inequalities of the form ax + b < c and ax + b > c. We will use the given inequality as an example to demonstrate the steps involved in solving linear inequalities.

Understanding the Given Inequality

The given inequality is -8 < 3x + 1 < 7. To solve this inequality, we need to isolate the variable x. We can start by subtracting 1 from all three parts of the inequality.

-8 < 3x + 1 - 1 < 7
-8 < 3x < 7 - 1
-8 < 3x < 6

Solving the Inequality

Next, we need to isolate the variable x. We can do this by dividing all three parts of the inequality by 3.

(-8)/3 < (3x)/3 < (6)/3
-8/3 < x < 2

Checking the Solution

To check our solution, we can substitute the values of x into the original inequality. Let's try x = -2.5.

-8 < 3(-2.5) + 1 < 7
-8 < -7.5 + 1 < 7
-8 < -6.5 < 7

Since -8 < -6.5 < 7 is true, our solution is correct.

Conclusion

In this article, we have demonstrated the steps involved in solving linear inequalities of the form ax + b < c and ax + b > c. We used the given inequality as an example to illustrate the process. By following these steps, you can solve linear inequalities and check your solutions.

Common Mistakes to Avoid

When solving linear inequalities, there are several common mistakes to avoid. Here are a few:

  • Not checking the solution: Always check your solution by substituting the values of x into the original inequality.
  • Not following the correct order of operations: When solving linear inequalities, it's essential to follow the correct order of operations (PEMDAS).
  • Not considering the direction of the inequality: When solving linear inequalities, it's essential to consider the direction of the inequality.

Real-World Applications

Linear inequalities have numerous real-world applications. Here are a few:

  • Finance: Linear inequalities are used to calculate interest rates, investment returns, and loan payments.
  • Science: Linear inequalities are used to model population growth, chemical reactions, and physical systems.
  • Engineering: Linear inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Practice Problems

Here are a few practice problems to help you reinforce your understanding of linear inequalities:

  • Solve the inequality 2x + 3 < 5.
  • Solve the inequality x - 2 > 3.
  • Solve the inequality 4x - 1 < 9.

Conclusion

In conclusion, linear inequalities are a fundamental concept in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve linear inequalities and check your solutions. Remember to avoid common mistakes, such as not checking the solution and not following the correct order of operations.

Final Answer

The final answer is:

-8/3 < x < 2

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of ax + b < c, where a, b, and c are constants, and x is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable x. You can do this by adding or subtracting the same value from all three parts of the inequality, and then dividing all three parts by the same non-zero value.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that can be written in the form of ax + b = c, where a, b, and c are constants, and x is the variable. A linear inequality, on the other hand, is an inequality that can be written in the form of ax + b < c or ax + b > c.

Q: How do I check my solution to a linear inequality?

A: To check your solution to a linear inequality, you need to substitute the values of x into the original inequality and verify that it is true.

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

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

  • Not checking the solution
  • Not following the correct order of operations (PEMDAS)
  • Not considering the direction of the inequality

Q: What are some real-world applications of linear inequalities?

A: Linear inequalities have numerous real-world applications, including:

  • Finance: Linear inequalities are used to calculate interest rates, investment returns, and loan payments.
  • Science: Linear inequalities are used to model population growth, chemical reactions, and physical systems.
  • Engineering: Linear inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: How do I solve a linear inequality with fractions?

A: To solve a linear inequality with fractions, you need to follow the same steps as solving a linear inequality with integers. However, you may need to multiply both sides of the inequality by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, you need to be careful when using a calculator to ensure that you are following the correct order of operations and that you are not making any mistakes.

Q: How do I graph a linear inequality?

A: To graph a linear inequality, you need to graph the corresponding linear equation and then shade the region that satisfies the inequality.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form of ax + b < c or ax + b > c, where a, b, and c are constants, and x is the variable. A quadratic inequality, on the other hand, is an inequality that can be written in the form of ax^2 + bx + c < d or ax^2 + bx + c > d, where a, b, c, and d are constants, and x is the variable.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution set.

Conclusion

In conclusion, linear inequalities are a fundamental concept in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve linear inequalities and check your solutions. Remember to avoid common mistakes, such as not checking the solution and not following the correct order of operations.

Final Answer

The final answer is:

-8/3 < x < 2

This is the solution to the given inequality.