Solve The Inequality: − 9.5 + 6 X ≤ 42.1 -9.5 + 6x \quad \leq \quad 42.1 − 9.5 + 6 X ≤ 42.1 Apply Properties To Solve:1. Add 9.5 To Both Sides.2. Divide By 6.(Reset To Start Over If Needed.)

Mar 13, 2025 by ADMIN 191 views

Solving Inequalities: A Step-by-Step Guide

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In mathematics, inequalities are an essential part of solving equations and understanding the relationships between variables. In this article, we will focus on solving a specific inequality using basic algebraic properties. We will break down the solution into manageable steps, making it easier to understand and apply to similar problems.

Understanding the Inequality

The given inequality is $-9.5 + 6x \quad \leq \quad 42.1$ . Our goal is to isolate the variable $x$ and find the range of values that satisfy the inequality.

What are Inequalities?

Inequalities are mathematical statements that compare two expressions using greater than, less than, greater than or equal to, or less than or equal to symbols. In this case, we have a less than or equal to symbol ( $\leq$ ), indicating that the expression on the left-hand side is less than or equal to the expression on the right-hand side.

Step 1: Add 9.5 to Both Sides

To solve the inequality, we need to isolate the variable $x$ . The first step is to add 9.5 to both sides of the inequality. This will help us get rid of the negative term on the left-hand side.

# Given inequality
inequality = "-9.5 + 6x <= 42.1"
new_inequality = "6x <= 42.1 + 9.5"

By adding 9.5 to both sides, we get:

6x \leq 51.6

Step 2: Divide by 6

Now that we have the inequality in the form $6x \leq 51.6$ , we need to isolate the variable $x$ . The next step is to divide both sides of the inequality by 6.

# Divide both sides by 6
new_inequality = "x <= 51.6 / 6"

By dividing both sides by 6, we get:

x \leq 8.6

Conclusion

In this article, we solved the inequality $-9.5 + 6x \quad \leq \quad 42.1$ using basic algebraic properties. We added 9.5 to both sides to get rid of the negative term, and then divided both sides by 6 to isolate the variable $x$ . The final solution is $x \leq 8.6$ , indicating that the value of $x$ must be less than or equal to 8.6.

Why is Solving Inequalities Important?

Solving inequalities is an essential part of mathematics, with applications in various fields such as physics, engineering, and economics. Inequalities help us understand the relationships between variables and make predictions about real-world phenomena. By mastering the skills of solving inequalities, you will be better equipped to tackle complex problems and make informed decisions.

Tips for Solving Inequalities

Here are some tips to help you solve inequalities like a pro:

Read the inequality carefully: Make sure you understand the inequality and what it's asking for.
Use basic algebraic properties: Add, subtract, multiply, and divide both sides of the inequality to isolate the variable.
Check your work: Verify that your solution satisfies the original inequality.
Practice, practice, practice: The more you practice solving inequalities, the more comfortable you'll become with the process.

Common Mistakes to Avoid

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

Not reading the inequality carefully: Make sure you understand the inequality and what it's asking for.
Not using basic algebraic properties: Don't forget to add, subtract, multiply, and divide both sides of the inequality to isolate the variable.
Not checking your work: Verify that your solution satisfies the original inequality.
Not practicing enough: The more you practice solving inequalities, the more comfortable you'll become with the process.

Conclusion

Solving inequalities is an essential part of mathematics, with applications in various fields such as physics, engineering, and economics. By mastering the skills of solving inequalities, you will be better equipped to tackle complex problems and make informed decisions. Remember to read the inequality carefully, use basic algebraic properties, check your work, and practice, practice, practice. With these tips and a little practice, you'll be solving inequalities like a pro in no time!

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

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two expressions using greater than, less than, greater than or equal to, or less than or equal to symbols.

Q: What are the different types of inequalities?

A: There are four main types of inequalities:

Greater than ( $>$ ): This inequality states that the expression on the left-hand side is greater than the expression on the right-hand side.
Less than ( $<$ ): This inequality states that the expression on the left-hand side is less than the expression on the right-hand side.
Greater than or equal to ( $\geq$ ): This inequality states that the expression on the left-hand side is greater than or equal to the expression on the right-hand side.
Less than or equal to ( $\leq$ ): This inequality states that the expression on the left-hand side is less than or equal to the expression on the right-hand side.

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.

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

A: Here are some common mistakes to avoid when solving inequalities:

Not reading the inequality carefully: Make sure you understand the inequality and what it's asking for.
Not using basic algebraic properties: Don't forget to add, subtract, multiply, and divide both sides of the inequality to isolate the variable.
Not checking your work: Verify that your solution satisfies the original inequality.
Not practicing enough: The more you practice solving inequalities, the more comfortable you'll become with the process.

Q: How do I check my work when solving inequalities?

A: To check your work, you need to verify that your solution satisfies the original inequality. You can do this by plugging your solution back into the original inequality and checking if it's true.

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

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

Physics: Inequalities are used to describe the relationships between physical quantities such as distance, time, and velocity.
Engineering: Inequalities are used to design and optimize systems such as bridges, buildings, and electronic circuits.
Economics: Inequalities are used to model economic systems and make predictions about future trends.
Computer Science: Inequalities are used to solve problems in computer science such as sorting and searching algorithms.

Q: How can I practice solving inequalities?

A: Here are some ways to practice solving inequalities:

Work on practice problems: You can find practice problems in textbooks, online resources, or by creating your own problems.
Use online resources: There are many online resources available that provide practice problems and solutions for inequalities.
Join a study group: Joining a study group can be a great way to practice solving inequalities with others.
Take online courses: There are many online courses available that teach inequalities and provide practice problems.

Q: What are some common inequalities that I should know?

A: Here are some common inequalities that you should know:

Linear inequalities: These are inequalities that can be written in the form $ax + b \leq c$ or $ax + b \geq c$ .
Quadratic inequalities: These are inequalities that can be written in the form $ax^2 + bx + c \leq 0$ or $ax^2 + bx + c \geq 0$ .
Rational inequalities: These are inequalities that can be written in the form $\frac{ax + b}{cx + d} \leq 0$ or $\frac{ax + b}{cx + d} \geq 0$ .