Solve The Inequality:$\[ \frac{20}{4} \ \textless \ \frac{x+4}{2} \ \textless \ \frac{25}{4} \\]Express The Solution In Interval Notation Using Decimal Form.

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve them. Specifically, we will focus on solving the given inequality: 204 \textless x+42 \textless 254\frac{20}{4} \ \textless \ \frac{x+4}{2} \ \textless \ \frac{25}{4}. We will break down the solution into manageable steps and express the final answer in interval notation using decimal form.

Understanding the Inequality

Before we dive into solving the inequality, let's understand what it means. The given inequality is a compound inequality, which means it consists of two separate inequalities joined by the word "and." In this case, we have:

204 \textless x+42\frac{20}{4} \ \textless \ \frac{x+4}{2} (Inequality 1)

and

x+42 \textless 254\frac{x+4}{2} \ \textless \ \frac{25}{4} (Inequality 2)

Solving Inequality 1

Let's start by solving Inequality 1. To do this, we need to isolate the variable x. We can start by multiplying both sides of the inequality by 2, which is the denominator of the fraction on the right-hand side.

204 \textless x+42\frac{20}{4} \ \textless \ \frac{x+4}{2}

Multiplying both sides by 2:

10 \textless x+410 \ \textless \ x+4

Subtracting 4 from both sides:

6 \textless x6 \ \textless \ x

Solving Inequality 2

Now, let's move on to solving Inequality 2. Again, we need to isolate the variable x. We can start by multiplying both sides of the inequality by 2, which is the denominator of the fraction on the right-hand side.

x+42 \textless 254\frac{x+4}{2} \ \textless \ \frac{25}{4}

Multiplying both sides by 2:

x+4 \textless 252x+4 \ \textless \ \frac{25}{2}

Subtracting 4 from both sides:

x \textless 252−4x \ \textless \ \frac{25}{2} - 4

Simplifying the right-hand side:

x \textless 212x \ \textless \ \frac{21}{2}

Combining the Solutions

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

From Inequality 1, we have:

6 \textless x6 \ \textless \ x

From Inequality 2, we have:

x \textless 212x \ \textless \ \frac{21}{2}

Combining the two inequalities, we get:

6 \textless x \textless 2126 \ \textless \ x \ \textless \ \frac{21}{2}

Expressing the Solution in Interval Notation

The final step is to express the solution in interval notation using decimal form. To do this, we need to convert the fractions to decimals.

212=10.5\frac{21}{2} = 10.5

So, the solution in interval notation is:

(6,10.5)(6, 10.5)

Conclusion

In this article, we solved the given inequality and expressed the solution in interval notation using decimal form. We broke down the solution into manageable steps and used algebraic manipulations to isolate the variable x. We also combined the solutions from both inequalities to find the final answer. We hope this article has provided a clear and concise guide to solving inequalities.

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Introduction

In our previous article, we solved the inequality 204 \textless x+42 \textless 254\frac{20}{4} \ \textless \ \frac{x+4}{2} \ \textless \ \frac{25}{4} and expressed the solution in interval notation using decimal form. In this article, we will provide a Q&A guide to help you better understand the solution and address any questions you may have.

Q: What is the difference between a compound inequality and a single inequality?

A: A compound inequality is a statement that combines two or more inequalities with the word "and" or "or." In the case of the given inequality, we have two separate inequalities joined by the word "and." A single inequality, on the other hand, is a statement that compares two expressions using a single inequality symbol (e.g., <, >, ≤, ≥).

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions. In the case of the given inequality, we solved each inequality separately and then combined the solutions to find the final answer.

Q: What is the purpose of multiplying both sides of an inequality by a constant?

A: Multiplying both sides of an inequality by a constant is a way to isolate the variable x. When you multiply both sides of an inequality by a positive constant, the direction of the inequality remains the same. However, when you multiply both sides of an inequality by a negative constant, the direction of the inequality is reversed.

Q: How do I express the solution in interval notation?

A: To express the solution in interval notation, you need to use parentheses or brackets to indicate the endpoints of the solution set. In the case of the given inequality, we expressed the solution in interval notation as (6, 10.5).

Q: What is the difference between a closed interval and an open interval?

A: A closed interval is an interval that includes the endpoints, while an open interval is an interval that does not include the endpoints. In the case of the given inequality, we expressed the solution as an open interval (6, 10.5), which means that the endpoints 6 and 10.5 are not included in the solution set.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. However, you need to be careful when using a calculator to solve an inequality, as the calculator may not always give you the correct answer. It's always a good idea to check your work by hand to make sure that the solution is correct.

Q: How do I check my work when solving an inequality?

A: To check your work when solving an inequality, you need to plug in a test value into the inequality and see if it is true or false. If the test value satisfies the inequality, then the solution is correct. If the test value does not satisfy the inequality, then the solution is incorrect.

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

A: Some common mistakes to avoid when solving an inequality include:

  • Not following the order of operations
  • Not multiplying both sides of the inequality by a constant
  • Not checking the direction of the inequality
  • Not expressing the solution in interval notation

Conclusion

In this article, we provided a Q&A guide to help you better understand the solution to the inequality 204 \textless x+42 \textless 254\frac{20}{4} \ \textless \ \frac{x+4}{2} \ \textless \ \frac{25}{4}. We addressed common questions and provided tips and tricks to help you solve inequalities with confidence. We hope this article has been helpful in your understanding of inequalities.