Which Represents The Solution Set Of $5(x+5) \ \textless \ 85$?A. $x \ \textless \ 12$ B. $ X \textgreater 12 X \ \textgreater \ 12 X \textgreater 12 [/tex] C. $x \ \textless \ 16$ D. $x \ \textgreater \ 16$

Introduction

In mathematics, inequalities are a fundamental concept that helps us compare values and make decisions. In this article, we will focus on solving linear inequalities, specifically the inequality $5(x+5) \ \textless \ 85$. We will break down the solution step by step and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is $5(x+5) \ \textless \ 85$. To solve this inequality, we need to isolate the variable x. The first step is to distribute the 5 to the terms inside the parentheses.

$$5(x+5) \ \textless \ 85$$

$$5x + 25 \ \textless \ 85$$

Simplifying the Inequality

Now that we have distributed the 5, we can simplify the inequality by subtracting 25 from both sides.

$$5x + 25 - 25 \ \textless \ 85 - 25$$

$$5x \ \textless \ 60$$

Isolating the Variable

The next step is to isolate the variable x by dividing both sides of the inequality by 5.

$$\frac{5x}{5} \ \textless \ \frac{60}{5}$$

$$x \ \textless \ 12$$

Analyzing the Solution

Now that we have isolated the variable x, we can analyze the solution. The inequality $x \ \textless \ 12$ means that x is less than 12. This is a strict inequality, which means that x cannot be equal to 12.

Comparing the Solutions

Let's compare the solution $x \ \textless \ 12$ with the given options.

A. $x \ \textless \ 12$ B. $x \ \textgreater \ 12$ C. $x \ \textless \ 16$ D. $x \ \textgreater \ 16$

The correct solution is option A, $x \ \textless \ 12$.

Conclusion

In this article, we solved the inequality $5(x+5) \ \textless \ 85$ step by step. We distributed the 5, simplified the inequality, isolated the variable x, and analyzed the solution. We also compared the solution with the given options and found that the correct solution is option A, $x \ \textless \ 12$.

Tips and Tricks

• When solving inequalities, always follow the order of operations (PEMDAS).
• When distributing a coefficient to terms inside parentheses, make sure to multiply each term by the coefficient.
• When simplifying an inequality, make sure to perform the same operation on both sides of the inequality.
• When isolating a variable, make sure to divide both sides of the inequality by the coefficient.

Practice Problems

1. Solve the inequality $3(x-2) \ \textless \ 21$.
2. Solve the inequality $2(x+1) \ \textgreater \ 16$.
3. Solve the inequality $x-5 \ \textless \ 10$.

Answer Key

1. $$x \ \textless \ 9$$

2. $$x \ \textgreater \ 7$$

3. x \ \textless \ 15$<br/>

Introduction

In our previous article, we solved the inequality $5(x+5) \ \textless \ 85$ step by step. We distributed the 5, simplified the inequality, isolated the variable x, and analyzed the solution. In this article, we will provide a Q&A guide to help you understand and solve inequalities.

Q: What is an inequality?

A: An inequality is a statement that compares two values using a mathematical symbol, such as <, >, ≤, or ≥.

Q: What are the different types of inequalities?

A: There are two main types of inequalities: linear inequalities and quadratic inequalities. Linear inequalities are inequalities that can be written in the form ax + b < c, where a, b, and c are constants. Quadratic inequalities are inequalities that involve a quadratic expression.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, follow these steps:

1. Distribute any coefficients to the terms inside parentheses.
2. Simplify the inequality by performing the same operation on both sides.
3. Isolate the variable by dividing both sides by the coefficient.
4. Analyze the solution and determine the correct inequality sign.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses a strict inequality sign, such as < or >. A non-strict inequality is an inequality that uses a non-strict inequality sign, such as ≤ or ≥.

Q: How do I know which inequality sign to use?

A: To determine which inequality sign to use, follow these steps:

1. Look at the inequality and determine the direction of the inequality.
2. If the inequality is pointing to the left, use the < or ≤ sign.
3. If the inequality is pointing to the right, use the > or ≥ sign.

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

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

1. Not distributing coefficients to terms inside parentheses.
2. Not simplifying the inequality by performing the same operation on both sides.
3. Not isolating the variable by dividing both sides by the coefficient.
4. Not analyzing the solution and determining the correct inequality sign.

Q: How do I practice solving inequalities?

A: To practice solving inequalities, try the following:

1. Start with simple inequalities and work your way up to more complex ones.
2. Use online resources, such as Khan Academy or Mathway, to practice solving inequalities.
3. Work with a partner or tutor to help you understand and solve inequalities.
4. Take practice quizzes or tests to assess your understanding of inequalities.

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

A: Inequalities have many real-world applications, including:

1. Finance: Inequalities are used to calculate interest rates and investment returns.
2. Science: Inequalities are used to model population growth and decay.
3. Engineering: Inequalities are used to design and optimize systems.
4. Economics: Inequalities are used to model economic systems and make predictions.

Conclusion

In this article, we provided a Q&A guide to help you understand and solve inequalities. We covered topics such as the definition of an inequality, the different types of inequalities, and common mistakes to avoid. We also provided tips and resources for practicing solving inequalities and real-world applications of inequalities.