Solve The Inequality: 5 ( A − 5 ) \textgreater 25 5(a-5)\ \textgreater \ 25 5 ( A − 5 ) \textgreater 25 A. A \textgreater 30 A \ \textgreater \ 30 A \textgreater 30 B. A \textless 30 A \ \textless \ 30 A \textless 30 C. A \textgreater 10 A \ \textgreater \ 10 A \textgreater 10 D. A \textgreater 5 A \ \textgreater \ 5 A \textgreater 5

Introduction

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, or equal to the other. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving the inequality $5(a-5) > 25$ and explore the different solution options.

Understanding the Inequality

The given inequality is $5(a-5) > 25$. To solve this inequality, we need to isolate the variable $a$ on one side of the inequality sign. The first step is to simplify the left-hand side of the inequality by distributing the coefficient $5$ to the terms inside the parentheses.

Distributing the Coefficient

When we distribute the coefficient $5$ to the terms inside the parentheses, we get:

$$5(a-5) = 5a - 25$$

Now, the inequality becomes:

$$5a - 25 > 25$$

Adding 25 to Both Sides

To isolate the term with the variable $a$, we need to add $25$ to both sides of the inequality. This will maintain the direction of the inequality.

$$5a - 25 + 25 > 25 + 25$$

Simplifying the left-hand side, we get:

$$5a > 50$$

Dividing Both Sides by 5

To solve for the variable $a$, we need to divide both sides of the inequality by $5$. This will isolate the variable on one side of the inequality sign.

$$\frac{5a}{5} > \frac{50}{5}$$

Simplifying the left-hand side, we get:

$$a > 10$$

Solution Options

Now that we have solved the inequality, we need to determine which of the given solution options is correct. Let's examine each option:

• Option A: $a > 30$
• Option B: $a < 30$
• Option C: $a > 10$
• Option D: $a > 5$

Based on our solution, we can see that the correct option is:

• Option C: $a > 10$

This is because the solution to the inequality $5(a-5) > 25$ is $a > 10$.

Conclusion

Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. By following the steps outlined in this article, we can solve the inequality $5(a-5) > 25$ and determine the correct solution option. Remember to always simplify the left-hand side of the inequality, add or subtract the same value to both sides, and divide both sides by the same non-zero value to isolate the variable.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid common mistakes that can lead to incorrect solutions. Here are some common mistakes to watch out for:

• Not simplifying the left-hand side of the inequality
• Not adding or subtracting the same value to both sides
• Not dividing both sides by the same non-zero value
• Not maintaining the direction of the inequality

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Practice Problems

To reinforce your understanding of solving inequalities, try the following practice problems:

• Solve the inequality $3(x+2) > 12$
• Solve the inequality $2(y-3) < 10$
• Solve the inequality $4(a-1) \geq 16$

Remember to follow the steps outlined in this article and avoid common mistakes to ensure accurate solutions.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields, including:

• Business: Inequality solutions can help businesses determine the optimal price for a product or service, based on factors such as demand and competition.
• Economics: Inequality solutions can help economists model and analyze economic systems, including the behavior of supply and demand.
• Science: Inequality solutions can help scientists model and analyze complex systems, including population growth and disease spread.

By understanding how to solve inequalities, you can apply this knowledge to real-world problems and make informed decisions.

Conclusion

Introduction

In our previous article, we explored the concept of solving inequalities and provided a step-by-step guide on how to solve the inequality $5(a-5) > 25$. In this article, we will address some of the most frequently asked questions about solving inequalities.

Q&A

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, indicating whether one is greater than, less than, or equal to the other.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities involve a single variable and a linear expression, while quadratic inequalities involve a single variable and a quadratic expression.

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, while maintaining the direction of the inequality. This can be done by adding or subtracting the same value to both sides, and dividing both sides by the same non-zero value.

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

A: A linear inequality involves a single variable and a linear expression, while a quadratic inequality involves a single variable and a quadratic expression. Quadratic inequalities are more complex and require additional steps to solve.

Q: How do I solve a quadratic inequality?

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

Q: What is the sign chart method?

A: The sign chart method is a technique used to determine the solution set of a quadratic inequality. It involves creating a chart with the critical points of the quadratic expression and then testing each interval to determine the sign of the expression.

Q: How do I determine the critical points of a quadratic expression?

A: The critical points of a quadratic expression are the values of the variable that make the expression equal to zero. To determine the critical points, you need to factor the quadratic expression, if possible, or use the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula to solve a quadratic inequality?

A: To use the quadratic formula to solve a quadratic inequality, you need to first solve the quadratic equation using the formula. Then, you need to test each interval to determine the sign of the expression.

Q: What is the solution set of a quadratic inequality?

A: The solution set of a quadratic inequality is the set of all values of the variable that satisfy the inequality. It can be represented graphically as a region on the number line.

Q: How do I graph the solution set of a quadratic inequality?

A: To graph the solution set of a quadratic inequality, you need to plot the critical points on the number line and then test each interval to determine the sign of the expression.

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

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

• Not simplifying the left-hand side of the inequality
• Not adding or subtracting the same value to both sides
• Not dividing both sides by the same non-zero value
• Not maintaining the direction of the inequality

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through example problems and exercises. You can also use online resources and practice tests to help you improve your skills.

Conclusion

Solving inequalities is a fundamental concept in mathematics that has numerous real-world applications. By understanding how to solve inequalities, you can apply this knowledge to real-world problems and make informed decisions. In this article, we addressed some of the most frequently asked questions about solving inequalities and provided a step-by-step guide on how to solve quadratic inequalities. Remember to avoid common mistakes and practice solving inequalities to reinforce your understanding. With this knowledge, you can apply inequality solutions to real-world problems and make informed decisions.