Reverse the inequality sign when multiplying by a negative number

Oct 13, 2025 by miodeya.com 46 views

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve a specific inequality: -1/5x + 3 > 9. Inequalities are mathematical expressions that contain a variable and a comparison operator, such as greater than or less than. Solving an inequality involves finding the values of the variable that make the inequality true. In this case, we will use algebraic techniques to isolate the variable x and find the solution set.

Understanding the Inequality

The given inequality is -1/5x + 3 > 9. To begin solving this inequality, we need to understand the properties of inequalities. Inequalities can be added, subtracted, multiplied, or divided by the same value, but the direction of the inequality sign may change depending on the operation. For example, if we multiply both sides of an inequality by a negative number, the direction of the inequality sign will reverse.

Isolating the Variable

To solve the inequality -1/5x + 3 > 9, we need to isolate the variable x. We can start by subtracting 3 from both sides of the inequality:

-1/5x + 3 - 3 > 9 - 3

This simplifies to:

-1/5x > 6

Multiplying by a Negative Number

To get rid of the fraction, we can multiply both sides of the inequality by -5. However, since we are multiplying by a negative number, we need to reverse the direction of the inequality sign:

(-5)(-1/5x) > (-5)(6)

This simplifies to:

x < -30

Checking the Solution

To ensure that our solution is correct, we need to check it by plugging it back into the original inequality. Let's substitute x = -30 into the original inequality:

-1/5(-30) + 3 > 9

This simplifies to:

6 + 3 > 9

Which is true. Therefore, our solution x < -30 is correct.

Conclusion

Solving the inequality -1/5x + 3 > 9 involves isolating the variable x and using algebraic techniques to find the solution set. By subtracting 3 from both sides and multiplying by a negative number, we were able to isolate x and find the solution x < -30. This solution was then checked by plugging it back into the original inequality, confirming that it is correct.

Tips and Tricks

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

Reverse the inequality sign when multiplying by a negative number: When multiplying both sides of an inequality by a negative number, the direction of the inequality sign will reverse.
Check the solution: To ensure that your solution is correct, plug it back into the original inequality and check if it's true.
Use algebraic techniques: Inequalities can be solved using algebraic techniques, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Common Mistakes

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

Not reversing the inequality sign when multiplying by a negative number: Failing to reverse the inequality sign when multiplying by a negative number can lead to incorrect solutions.
Not checking the solution: Failing to check the solution can lead to incorrect solutions.
Not using algebraic techniques: Failing to use algebraic techniques can make it difficult to solve inequalities.

Real-World Applications

Inequalities have many real-world applications. Here are a few examples:

Finance: Inequalities are used in finance to calculate interest rates, investment returns, and loan payments.
Science: Inequalities are used in science to model population growth, chemical reactions, and physical systems.
Engineering: Inequalities are used in engineering to design and optimize systems, such as bridges, buildings, and electronic circuits.

Conclusion

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Introduction

In the previous article, we learned how to solve the inequality -1/5x + 3 > 9. In this article, we will answer some frequently asked questions (FAQs) about solving inequalities. Whether you're a student, teacher, or simply someone interested in mathematics, this article will provide you with a comprehensive guide to solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that contains a variable and a comparison operator, such as greater than or less than. Inequalities can be used to describe relationships between variables and can be used to model real-world situations.

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, subtracting, multiplying, or dividing both sides of the inequality by the same value. However, if you multiply or divide both sides by a negative number, you need to reverse the direction of the inequality sign.

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

A: An equation is a mathematical expression that contains a variable and an equal sign. Inequalities, on the other hand, contain a variable and a comparison operator, such as greater than or less than. While equations can be used to describe relationships between variables, inequalities can be used to describe relationships between variables and can be used to model real-world situations.

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

A: To check your solution to an inequality, you need to plug it back into the original inequality and check if it's true. If the solution satisfies the inequality, then it's correct. If not, then you need to re-solve the inequality.

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

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

Not reversing the inequality sign when multiplying or dividing by a negative number
Not checking the solution
Not using algebraic techniques to isolate the variable

Q: How do I use algebraic techniques to solve inequalities?

A: Algebraic techniques can be used to solve inequalities by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value. However, if you multiply or divide both sides by a negative number, you need to reverse the direction of the inequality sign.

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

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

Finance: Inequalities are used in finance to calculate interest rates, investment returns, and loan payments.
Science: Inequalities are used in science to model population growth, chemical reactions, and physical systems.
Engineering: Inequalities are used in engineering to design and optimize systems, such as bridges, buildings, and electronic circuits.

Q: How do I graph an inequality?

A: To graph an inequality, you need to plot the boundary line and shade the region that satisfies the inequality. If the inequality is of the form x > a, then you need to shade the region to the right of the boundary line. If the inequality is of the form x < a, then you need to shade the region to the left of the boundary line.

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 and then use the sign of the quadratic expression to determine the solution set. You can also use the quadratic formula to solve the inequality.

Conclusion

Solving inequalities is an essential skill in mathematics and has many real-world applications. By understanding how to solve inequalities, you can model real-world situations and make informed decisions. In this article, we answered some frequently asked questions about solving inequalities and provided a comprehensive guide to solving inequalities. Whether you're a student, teacher, or simply someone interested in mathematics, this article will provide you with the knowledge and skills you need to solve inequalities with confidence.