Solve The Double Inequality: $ 17 \leq 5x - 2 \ \textless \ 33 $

by ADMIN 67 views

Introduction

In mathematics, inequalities are used to represent relationships between variables. A double inequality is a statement that involves two inequalities joined by the word "and." In this article, we will focus on solving the double inequality $ 17 \leq 5x - 2 \ \textless \ 33 $. This type of problem requires us to find the values of x that satisfy both inequalities simultaneously.

Understanding the Double Inequality

The given double inequality is $ 17 \leq 5x - 2 \ \textless \ 33 $. This can be broken down into two separate inequalities:

  • $ 17 \leq 5x - 2 $
  • $ 5x - 2 \ \textless \ 33 $

Step 1: Solve the First Inequality

To solve the first inequality, we need to isolate the variable x. We can start by adding 2 to both sides of the inequality:

$ 17 + 2 \leq 5x - 2 + 2 $

This simplifies to:

$ 19 \leq 5x $

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

$ \frac{19}{5} \leq x $

So, the solution to the first inequality is $ x \geq \frac{19}{5} $.

Step 2: Solve the Second Inequality

To solve the second inequality, we need to isolate the variable x. We can start by adding 2 to both sides of the inequality:

$ 5x - 2 + 2 \ \textless \ 33 + 2 $

This simplifies to:

$ 5x \ \textless \ 35 $

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

$ x \ \textless \ 7 $

So, the solution to the second inequality is $ x \ \textless \ 7 $.

Combining the Solutions

Now that we have solved both inequalities, we need to combine the solutions to find the values of x that satisfy both inequalities simultaneously. We can do this by finding the intersection of the two solution sets.

The solution to the first inequality is $ x \geq \frac{19}{5} $, and the solution to the second inequality is $ x \ \textless \ 7 $. To find the intersection of these two solution sets, we need to find the values of x that satisfy both inequalities.

Since $ \frac{19}{5} \approx 3.8 $, we can see that the solution to the first inequality is $ x \geq 3.8 $. Similarly, since $ 7 \approx 7.0 $, we can see that the solution to the second inequality is $ x \ \textless \ 7.0 $.

Therefore, the intersection of the two solution sets is $ 3.8 \leq x \ \textless \ 7.0 $.

Conclusion

In this article, we solved the double inequality $ 17 \leq 5x - 2 \ \textless \ 33 $. We broke down the double inequality into two separate inequalities and solved each one separately. We then combined the solutions to find the values of x that satisfy both inequalities simultaneously. The final solution is $ 3.8 \leq x \ \textless \ 7.0 $.

Key Takeaways

  • A double inequality is a statement that involves two inequalities joined by the word "and."
  • To solve a double inequality, we need to solve each inequality separately and then combine the solutions.
  • The solution to a double inequality is the intersection of the solution sets of the two inequalities.

Frequently Asked Questions

  • What is a double inequality? A double inequality is a statement that involves two inequalities joined by the word "and."
  • How do I solve a double inequality? To solve a double inequality, we need to solve each inequality separately and then combine the solutions.
  • What is the solution to the double inequality $ 17 \leq 5x - 2 \ \textless \ 33 $? The solution to the double inequality $ 17 \leq 5x - 2 \ \textless \ 33 $ is $ 3.8 \leq x \ \textless \ 7.0 $.
    Solve the Double Inequality: $ 17 \leq 5x - 2 \ \textless \ 33 $ ===========================================================

Q&A: Solving Double Inequalities

Q: What is a double inequality?

A: A double inequality is a statement that involves two inequalities joined by the word "and." It is a mathematical expression that represents a relationship between two variables, where one variable is greater than or equal to a certain value, and the other variable is less than a certain value.

Q: How do I solve a double inequality?

A: To solve a double inequality, you need to solve each inequality separately and then combine the solutions. This involves isolating the variable in each inequality and then finding the intersection of the solution sets.

Q: What is the solution to the double inequality $ 17 \leq 5x - 2 \ \textless \ 33 $?

A: The solution to the double inequality $ 17 \leq 5x - 2 \ \textless \ 33 $ is $ 3.8 \leq x \ \textless \ 7.0 $. This means that the value of x must be greater than or equal to 3.8 and less than 7.0.

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

A: To find the intersection of the solution sets, you need to find the values of x that satisfy both inequalities simultaneously. This involves finding the values of x that are greater than or equal to the lower bound of the first inequality and less than the upper bound of the second inequality.

Q: What if the solution sets do not intersect?

A: If the solution sets do not intersect, then there is no value of x that satisfies both inequalities simultaneously. In this case, the double inequality has no solution.

Q: Can I use algebraic methods to solve double inequalities?

A: Yes, you can use algebraic methods to solve double inequalities. This involves using algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable in each inequality.

Q: Can I use graphical methods to solve double inequalities?

A: Yes, you can use graphical methods to solve double inequalities. This involves graphing the two inequalities on a number line and finding the intersection of the two solution sets.

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

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

  • Not isolating the variable in each inequality
  • Not finding the intersection of the solution sets
  • Not considering the direction of the inequalities
  • Not checking for extraneous solutions

Q: How can I practice solving double inequalities?

A: You can practice solving double inequalities by working through examples and exercises in a textbook or online resource. You can also try solving double inequalities on your own and then checking your answers with a calculator or online tool.

Conclusion

Solving double inequalities requires a combination of algebraic and graphical methods. By understanding the concept of double inequalities and how to solve them, you can improve your problem-solving skills and become more confident in your ability to tackle complex mathematical problems.

Key Takeaways

  • A double inequality is a statement that involves two inequalities joined by the word "and."
  • To solve a double inequality, you need to solve each inequality separately and then combine the solutions.
  • The solution to a double inequality is the intersection of the solution sets of the two inequalities.
  • You can use algebraic and graphical methods to solve double inequalities.
  • Some common mistakes to avoid when solving double inequalities include not isolating the variable, not finding the intersection of the solution sets, and not considering the direction of the inequalities.