Solve The Following Inequality Using The Algebraic Approach: X + 2 \textgreater 3 X − 8 X + 2 \ \textgreater \ 3x - 8 X + 2 \textgreater 3 X − 8 A. X \textgreater − 5 X \ \textgreater \ -5 X \textgreater − 5 B. X \textless − 5 X \ \textless \ -5 X \textless − 5 C. X \textless 5 X \ \textless \ 5 X \textless 5 D. X \textgreater 5 X \ \textgreater \ 5 X \textgreater 5

Introduction

Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. 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 $x + 2 > 3x - 8$ using the algebraic approach.

Understanding the Inequality

The given inequality is $x + 2 > 3x - 8$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to simplify the inequality by combining like terms.

Simplifying the Inequality

To simplify the inequality, we can start by subtracting $x$ from both sides of the inequality. This will help us to isolate the variable $x$.

x + 2 > 3x - 8
x - x + 2 > 3x - x - 8
2 > 2x - 8

Next, we can add $8$ to both sides of the inequality to further simplify it.

2 > 2x - 8
2 + 8 > 2x - 8 + 8
10 > 2x

Isolating the Variable

Now that we have simplified the inequality, we can isolate the variable $x$ by dividing both sides of the inequality by $2$.

10 > 2x
10/2 > (2x)/2
5 > x

Writing the Solution

The final step is to write the solution to the inequality. Since we have isolated the variable $x$, we can write the solution as $x < 5$.

Conclusion

In this article, we have solved the inequality $x + 2 > 3x - 8$ using the algebraic approach. We started by simplifying the inequality, then isolated the variable $x$, and finally wrote the solution as $x < 5$. This solution indicates that the value of $x$ must be less than $5$ in order to satisfy the inequality.

Answer Options

Now that we have solved the inequality, let's compare our solution to the answer options provided.

• $x > -5$
• $x < -5$
• $x < 5$
• $x > 5$

Based on our solution, we can see that the correct answer is option c: $x < 5$.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

• Always simplify the inequality by combining like terms.
• Isolate the variable on one side of the inequality sign.
• Maintain the direction of the inequality.
• Check your solution by plugging in values to ensure that it satisfies the inequality.

By following these tips and tricks, you can become proficient in solving inequalities and apply this knowledge to a wide range of mathematical problems.

Common Mistakes

When solving inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

• Failing to simplify the inequality.
• Losing track of the direction of the inequality.
• Not isolating the variable on one side of the inequality sign.
• Not checking the solution by plugging in values.

By being aware of these common mistakes, you can avoid them and ensure that your solutions are accurate and reliable.

Real-World Applications

Solving inequalities has numerous real-world applications. Here are a few examples:

• In finance, inequalities are used to calculate interest rates and investment returns.
• In engineering, inequalities are used to design and optimize systems.
• In science, inequalities are used to model and analyze complex systems.

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

Conclusion

Introduction

In our previous article, we explored the basics of solving inequalities using the algebraic approach. In this article, we will delve deeper into the world of inequalities and answer some of the most frequently asked questions.

Q&A

Q: What is an inequality?

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

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, while maintaining the direction of the inequality. You can do this by simplifying the inequality, combining like terms, and using inverse operations.

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

A: A linear inequality is an inequality that can be written in the form $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

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 of the quadratic expression to determine the solution set. You can also use the quadratic formula to find the roots of the quadratic equation and then use a number line or a graph to determine the solution set.

Q: What is the concept of a number line?

A: A number line is a visual representation of the real number system, with numbers arranged in a straight line. It is a useful tool for solving inequalities and graphing functions.

Q: How do I use a number line to solve an inequality?

A: To use a number line to solve an inequality, you need to plot the solution set on the number line and then determine the interval that satisfies the inequality.

Q: What is the concept of a graph?

A: A graph is a visual representation of a function or an inequality. It is a useful tool for solving inequalities and graphing functions.

Q: How do I use a graph to solve an inequality?

A: To use a graph to solve an inequality, you need to plot the solution set on the graph and then determine the interval that satisfies the inequality.

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

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

• Failing to simplify the inequality
• Losing track of the direction of the inequality
• Not isolating the variable on one side of the inequality sign
• Not checking the solution by plugging in values

Q: How do I check my solution by plugging in values?

A: To check your solution by plugging in values, you need to substitute a value from the solution set into the original inequality and then determine if the inequality is true or false.

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

A: Some real-world applications of solving inequalities include:

• In finance, inequalities are used to calculate interest rates and investment returns.
• In engineering, inequalities are used to design and optimize systems.
• In science, inequalities are used to model and analyze complex systems.

Conclusion

In conclusion, solving inequalities is a fundamental skill that has numerous applications in mathematics and real-world problems. By understanding the basics of solving inequalities and using the concepts of number lines and graphs, you can become proficient in solving inequalities and apply this knowledge to a wide range of mathematical problems. Remember to simplify the inequality, isolate the variable, and maintain the direction of the inequality. With practice and patience, you can become a master of solving inequalities and tackle even the most complex problems with confidence.

Additional Resources

For more information on solving inequalities, check out the following resources:

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Practice Problems

Try solving the following inequalities:

• $x + 3 > 2x - 2$
• $x^2 + 4x + 4 > 0$
• $x - 2 < 3x + 1$

Answer Key

• $x > -1$
• $x > -2$
• $x < 5$