Solve The Inequality: 9 ( X − 7 ) \textless − 90 9(x-7)\ \textless \ -90 9 ( X − 7 ) \textless − 90

by ADMIN 101 views

Introduction

Inequalities are mathematical expressions that compare two values, often with a less-than or greater-than symbol. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality 9(x−7)<−909(x-7) < -90. We will break down the solution into manageable steps, using algebraic manipulations and logical reasoning.

Step 1: Distribute the Coefficient

The first step in solving the inequality is to distribute the coefficient 99 to the terms inside the parentheses. This will allow us to simplify the expression and make it easier to work with.

9(x−7)<−909(x-7) < -90

Distributing the coefficient 99 to the terms inside the parentheses, we get:

9x−63<−909x - 63 < -90

Step 2: Add 63 to Both Sides

The next step is to add 6363 to both sides of the inequality. This will help us isolate the term with the variable xx.

9x−63+63<−90+639x - 63 + 63 < -90 + 63

Simplifying the expression, we get:

9x<−279x < -27

Step 3: Divide Both Sides by 9

Now that we have isolated the term with the variable xx, we can divide both sides of the inequality by 99. This will give us the final solution to the inequality.

9x9<−279\frac{9x}{9} < \frac{-27}{9}

Simplifying the expression, we get:

x<−3x < -3

Conclusion

In this article, we solved the inequality 9(x−7)<−909(x-7) < -90 using algebraic manipulations and logical reasoning. We broke down the solution into manageable steps, distributing the coefficient, adding 6363 to both sides, and dividing both sides by 99. The final solution to the inequality is x<−3x < -3. This means that any value of xx that is less than −3-3 will satisfy the inequality.

Why is Solving Inequalities Important?

Solving inequalities is an essential skill in mathematics, with applications in various fields such as physics, engineering, and economics. Inequalities are used to model real-world problems, such as optimizing functions, finding maximum and minimum values, and making decisions based on uncertain data. By solving inequalities, we can gain a deeper understanding of the relationships between variables and make informed decisions.

Real-World Applications of Inequalities

Inequalities have numerous real-world applications, including:

  • Optimization: Inequalities are used to optimize functions, such as minimizing costs or maximizing profits.
  • Physics: Inequalities are used to model physical systems, such as motion, energy, and momentum.
  • Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Economics: Inequalities are used to model economic systems, such as supply and demand, inflation, and unemployment.

Tips and Tricks for Solving Inequalities

Here are some tips and tricks for solving inequalities:

  • Use algebraic manipulations: Inequalities can be solved using algebraic manipulations, such as adding, subtracting, multiplying, and dividing both sides of the inequality.
  • Use logical reasoning: Inequalities can be solved using logical reasoning, such as analyzing the relationships between variables and making informed decisions.
  • Check your work: Always check your work by plugging in values to ensure that the solution is correct.

Conclusion

Introduction

In our previous article, we solved the inequality 9(x−7)<−909(x-7) < -90 using algebraic manipulations and logical reasoning. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often with a less-than or greater-than symbol.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. This can be done using algebraic manipulations, such as adding, subtracting, multiplying, and dividing both sides of the inequality.

Q: What are some common types of inequalities?

A: There are several types of inequalities, including:

  • Linear inequalities: These are inequalities that can be written in the form ax+b<cax + b < c or ax+b>cax + b > c.
  • Quadratic inequalities: These are inequalities that can be written in the form ax2+bx+c<dax^2 + bx + c < d or ax2+bx+c>dax^2 + bx + c > d.
  • Absolute value inequalities: These are inequalities that involve absolute values, such as ∣x∣<a|x| < a or ∣x∣>a|x| > a.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. This can be done using algebraic manipulations, such as adding, subtracting, multiplying, and dividing both sides of the inequality.

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.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is positive, and when the expression inside the absolute value is negative.

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

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

  • Not checking the direction of the inequality: Make sure to check the direction of the inequality sign before solving the inequality.
  • Not considering all possible cases: Make sure to consider all possible cases when solving an inequality, such as when the expression inside the absolute value is positive or negative.
  • Not checking the solution: Make sure to check the solution by plugging in values to ensure that it is correct.

Q: How do I check my work when solving inequalities?

A: To check your work when solving inequalities, you need to plug in values to ensure that the solution is correct. This can be done by substituting values into the inequality and checking if the inequality is true or false.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics, with applications in various fields such as physics, engineering, and economics. By understanding the concepts and techniques involved in solving inequalities, you can solve a wide range of inequalities and gain a deeper understanding of the relationships between variables. Remember to check your work and consider all possible cases when solving inequalities.

Additional Resources

For additional resources on solving inequalities, including videos, tutorials, and practice problems, please visit the following websites:

  • Khan Academy: Khan Academy offers a comprehensive course on solving inequalities, including video tutorials and practice problems.
  • Mathway: Mathway is an online math problem solver that can help you solve inequalities and other math problems.
  • Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve inequalities and other math problems.

Final Tips

Here are some final tips for solving inequalities:

  • Practice, practice, practice: The more you practice solving inequalities, the more comfortable you will become with the concepts and techniques involved.
  • Use algebraic manipulations: Algebraic manipulations are a powerful tool for solving inequalities.
  • Check your work: Always check your work by plugging in values to ensure that the solution is correct.