Not reversing the inequality sign

Oct 13, 2025 by recoveryshake.com 72 views

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that two mathematical expressions are not equal, but one is either greater than or less than the other. In this article, we will focus on solving the inequality $-5u + 2(u - 4) \geq -23$ and simplify our answer as much as possible.

Understanding the Inequality

The given inequality is $-5u + 2(u - 4) \geq -23$ . To solve this inequality, we need to isolate the variable $u$ on one side of the inequality sign. The first step is to simplify the left-hand side of the inequality by combining like terms.

Simplifying the Left-Hand Side

To simplify the left-hand side, we need to distribute the $2$ to the terms inside the parentheses: $2(u - 4) = 2u - 8$ . Now, we can rewrite the inequality as $-5u + 2u - 8 \geq -23$ .

Combining Like Terms

Next, we need to combine the like terms on the left-hand side of the inequality. The like terms are $-5u$ and $2u$ , which can be combined as $-3u$ . Therefore, the inequality becomes $-3u - 8 \geq -23$ .

Isolating the Variable

Now, we need to isolate the variable $u$ on one side of the inequality sign. To do this, we can add $8$ to both sides of the inequality: $-3u - 8 + 8 \geq -23 + 8$ . This simplifies to $-3u \geq -15$ .

Dividing Both Sides

Finally, we need to divide both sides of the inequality by $-3$ to isolate the variable $u$ . However, when we divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign. Therefore, the inequality becomes $u \leq 5$ .

Conclusion

In conclusion, the solution to the inequality $-5u + 2(u - 4) \geq -23$ is $u \leq 5$ . This means that any value of $u$ that is less than or equal to $5$ satisfies the inequality.

Tips and Tricks

When solving inequalities, it's essential to follow the order of operations (PEMDAS) and simplify the left-hand side by combining like terms.
When dividing both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.
It's crucial to check the solution by plugging in a value of $u$ that satisfies the inequality to ensure that it's true.

Real-World Applications

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

Finance: Inequality can be used to model financial situations, such as determining the maximum amount of money that can be borrowed.
Science: Inequality can be used to model scientific phenomena, such as the relationship between variables in a scientific experiment.
Engineering: Inequality can be used to model engineering problems, such as determining the maximum stress on a material.

Common Mistakes to Avoid

Not simplifying the left-hand side: Failing to simplify the left-hand side of the inequality can lead to incorrect solutions.
Not reversing the inequality sign: Failing to reverse the inequality sign when dividing both sides by a negative number can lead to incorrect solutions.
Not checking the solution: Failing to check the solution by plugging in a value of $u$ that satisfies the inequality can lead to incorrect solutions.

Final Thoughts

Solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can simplify the inequality $-5u + 2(u - 4) \geq -23$ and find the solution. Remember to always simplify the left-hand side, reverse the inequality sign when dividing both sides by a negative number, and check the solution by plugging in a value of $u$ that satisfies the inequality.

Introduction

In our previous article, we discussed how to solve the inequality $-5u + 2(u - 4) \geq -23$ and simplify our answer as much as possible. In this article, we will provide a Q&A guide to help you better understand the concept of solving inequalities and address any questions or concerns you may have.

Q&A

Q: What is an inequality?

A: An inequality is a statement that two mathematical expressions are not equal, but one is either greater than or less than the other.

Q: What is the difference between an inequality and an equation?

A: An equation is a statement that two mathematical expressions are equal, while an inequality is a statement that two mathematical expressions are not equal.

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. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the order of operations when solving an inequality?

A: The order of operations when solving an inequality is the same as when solving an equation: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Q: What happens when I divide both sides of an inequality by a negative number?

A: When you divide both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you can plug in a value of the variable that satisfies the inequality and see if it's true.

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, not reversing the inequality sign when dividing both sides by a negative number, and not checking the solution.

Q: How do I apply inequalities to real-world problems?

A: Inequalities can be used to model real-world problems in various fields, including finance, science, and engineering. For example, you can use inequalities to determine the maximum amount of money that can be borrowed, or to model the relationship between variables in a scientific experiment.

Q: What are some examples of inequalities in real-world problems?

A: Some examples of inequalities in real-world problems include:

Determining the maximum amount of money that can be borrowed based on a person's income
Modeling the relationship between the amount of fertilizer used and the yield of a crop
Determining the maximum stress on a material based on its strength and the force applied to it

Conclusion

Solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can solve inequalities and apply them to real-world problems. Remember to always simplify the left-hand side, reverse the inequality sign when dividing both sides by a negative number, and check the solution by plugging in a value of the variable that satisfies the inequality.

Tips and Tricks

Practice solving inequalities with different types of problems to become more confident and proficient.
Use real-world examples to illustrate the concept of inequalities and make it more relatable.
Encourage students to think critically and creatively when applying inequalities to real-world problems.

Common Misconceptions

Some students may think that inequalities are only used in mathematics and have no real-world applications.
Some students may think that inequalities are only used to solve problems with two variables.
Some students may think that inequalities are only used in specific fields, such as finance or science.