Solve The Inequality:$\[ -3x + 6 \geq 0 \\]

$$

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more expressions. Solving inequalities is an essential skill that is used in various fields, including algebra, calculus, and statistics. In this article, we will focus on solving the inequality $-3x + 6 \geq 0$. We will break down the solution into step-by-step instructions, making it easy to understand and follow.

Understanding the Inequality

The given inequality is $-3x + 6 \geq 0$. This means that the expression $-3x + 6$ is greater than or equal to zero. To solve this inequality, we need to isolate the variable $x$.

Step 1: Subtract 6 from Both Sides

The first step in solving the inequality is to subtract 6 from both sides. This will help us isolate the term containing the variable $x$.

$$-3x + 6 \geq 0$$

Subtracting 6 from both sides gives us:

$$-3x \geq -6$$

Step 2: Divide Both Sides by -3

Next, we need to divide both sides of the inequality by -3. However, when we divide or multiply an inequality by a negative number, we need to reverse the direction of the inequality sign.

$$-3x \geq -6$$

Dividing both sides by -3 gives us:

$$x \leq 2$$

Step 3: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. The solution $x \leq 2$ can be written as $(-\infty, 2]$.

Conclusion

Solving the inequality $-3x + 6 \geq 0$ involves several steps, including subtracting 6 from both sides, dividing both sides by -3, and writing the solution in interval notation. By following these steps, we can find the solution to the inequality and express it in a clear and concise manner.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid. These include:

• Not reversing the direction of the inequality sign when dividing or multiplying by a negative number
• Not checking the solution by plugging in values
• Not writing the solution in interval notation

Tips and Tricks

Here are some tips and tricks to help you solve inequalities:

• Use a number line to visualize the solution
• Check the solution by plugging in values
• Use interval notation to express the solution

Real-World Applications

Solving inequalities has many real-world applications, including:

• Finance: In finance, inequalities are used to model the growth of investments and the impact of interest rates on savings.
• Science: In science, inequalities are used to model the behavior of physical systems and the impact of variables on outcomes.
• Engineering: In engineering, inequalities are used to design and optimize systems, including electrical circuits and mechanical systems.

Conclusion

$$

Introduction

In our previous article, we discussed how to solve the inequality $-3x + 6 \geq 0$. In this article, we will provide a Q&A guide to help you understand and apply the concepts of solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two or more expressions. It can be written in the form of $a \geq b$, $a \leq b$, or $a > b$, where $a$ and $b$ are expressions.

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, or by multiplying or dividing both sides by the same non-zero value.

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 of $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form of $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can follow these steps:

1. Subtract the constant term from both sides.
2. Divide both sides by the coefficient of the variable.
3. Write the solution in interval notation.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can follow these steps:

1. Factor the quadratic expression.
2. Set each factor equal to zero and solve for the variable.
3. Use a number line to determine the intervals where the inequality is true.
4. Write the solution in interval notation.

Q: What is the importance of checking the solution?

A: Checking the solution is an essential step in solving inequalities. It helps you to verify that the solution is correct and to ensure that you have not made any mistakes.

Q: How do I check the solution?

A: To check the solution, you can plug in values from each interval into the original inequality. If the inequality is true for all values in the interval, then the solution is correct.

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

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

• Not reversing the direction of the inequality sign when dividing or multiplying by a negative number.
• Not checking the solution by plugging in values.
• Not writing the solution in interval notation.

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

A: Inequalities have many real-world applications, including finance, science, and engineering. You can use inequalities to model the growth of investments, the behavior of physical systems, and the impact of variables on outcomes.

Conclusion

Solving inequalities is an essential skill that is used in various fields, including algebra, calculus, and statistics. By following the steps outlined in this article, you can solve inequalities and apply them to real-world problems. Remember to check the solution, avoid common mistakes, and use inequalities to model real-world situations.

Frequently Asked Questions

• Q: What is the difference between an inequality and an equation? A: An inequality is a mathematical statement that compares two or more expressions, while an equation is a mathematical statement that states that two expressions are equal.
• Q: How do I solve a system of inequalities? A: To solve a system of inequalities, you need to find the intersection of the solutions to each inequality.
• Q: What is the importance of using interval notation? A: Using interval notation helps you to express the solution in a clear and concise manner, and to visualize the solution on a number line.

Additional Resources

• Online Resources: There are many online resources available that can help you to learn and practice solving inequalities, including Khan Academy, Mathway, and Wolfram Alpha.
• Textbooks: There are many textbooks available that can help you to learn and practice solving inequalities, including "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by Michael Spivak.
• Practice Problems: There are many practice problems available that can help you to practice solving inequalities, including those found in textbooks and online resources.