Solve For $q$.$-3q \geq \frac{3}{4}$

$$$$

Introduction

In mathematics, solving inequalities is a crucial concept that helps us understand the relationship between different variables. In this article, we will focus on solving the inequality $-3q \geq \frac{3}{4}$, where $q$ is the variable we need to solve for. We will use step-by-step instructions and provide examples to help you understand the concept better.

Understanding the Inequality

The given inequality is $-3q \geq \frac{3}{4}$. To solve for $q$, we need to isolate the variable $q$ on one side of the inequality. The inequality states that the product of $-3$ and $q$ is greater than or equal to $\frac{3}{4}$.

Step 1: Multiply Both Sides by -3

To isolate $q$, we need to get rid of the coefficient $-3$ that is multiplied with $q$. We can do this by multiplying both sides of the inequality by $-3$. However, when we multiply or divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign.

-3q \geq \frac{3}{4}

Multiplying both sides by $-3$ gives us:

q \leq -\frac{3}{4} \times \frac{1}{-3}

Step 2: Simplify the Right-Hand Side

Now, we need to simplify the right-hand side of the inequality. We can do this by multiplying the numerator and denominator of the fraction by $-1$.

q \leq -\frac{3}{4} \times \frac{1}{-3}

Simplifying the right-hand side gives us:

q \leq \frac{3}{4} \times \frac{1}{3}

Step 3: Simplify the Fraction

Now, we need to simplify the fraction on the right-hand side of the inequality. We can do this by dividing the numerator and denominator by their greatest common divisor, which is $1$.

q \leq \frac{3}{4} \times \frac{1}{3}

Simplifying the fraction gives us:

q \leq \frac{1}{4}

Conclusion

In this article, we solved the inequality $-3q \geq \frac{3}{4}$ by multiplying both sides by $-3$ and simplifying the right-hand side. We found that the solution to the inequality is $q \leq \frac{1}{4}$. This means that the value of $q$ must be less than or equal to $\frac{1}{4}$.

Examples

Here are some examples of how to use the solution to the inequality:

• If $q = \frac{1}{4}$, then the inequality is true.
• If $q = \frac{1}{2}$, then the inequality is false.
• If $q = 0$, then the inequality is true.

Tips and Tricks

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

• Always read the inequality carefully and understand what it is saying.
• Use inverse operations to isolate the variable on one side of the inequality.
• Be careful when multiplying or dividing both sides of an inequality by a negative number.
• Simplify the right-hand side of the inequality by dividing the numerator and denominator by their greatest common divisor.

Common Mistakes

Here are some common mistakes to avoid when solving inequalities like this one:

• Not reading the inequality carefully and understanding what it is saying.
• Not using inverse operations to isolate the variable on one side of the inequality.
• Not being careful when multiplying or dividing both sides of an inequality by a negative number.
• Not simplifying the right-hand side of the inequality by dividing the numerator and denominator by their greatest common divisor.

Final Answer

The final answer to the inequality $-3q \geq \frac{3}{4}$ is $q \leq \frac{1}{4}$.

$$$$

Introduction

In our previous article, we solved the inequality $-3q \geq \frac{3}{4}$ and found that the solution is $q \leq \frac{1}{4}$. In this article, we will answer some frequently asked questions about solving inequalities like this one.

Q&A

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to read the inequality carefully and understand what it is saying. This includes identifying the variable, the coefficient, and the constant term.

Q: How do I isolate the variable on one side of the inequality?

A: To isolate the variable on one side of the inequality, you need to use inverse operations. For example, if the variable is multiplied by a coefficient, you can divide both sides of the inequality by that coefficient to isolate the variable.

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

A: When you multiply or divide both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign. For example, if the inequality is $a \geq b$ and you multiply both sides by $-1$, the inequality becomes $-a \leq -b$.

Q: How do I simplify the right-hand side of the inequality?

A: To simplify the right-hand side of the inequality, you need to divide the numerator and denominator by their greatest common divisor. This will give you the simplest form of the fraction.

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

A: An inequality is a statement that two expressions are not equal, while an equation is a statement that two expressions are equal. For example, $x > 2$ is an inequality, while $x = 2$ is an equation.

Q: Can I use the same steps to solve a linear inequality as I would to solve a linear equation?

A: Yes, you can use the same steps to solve a linear inequality as you would to solve a linear equation. However, you need to be careful when multiplying or dividing both sides of the inequality by a negative number, as this will reverse the direction of the inequality sign.

Q: What is the solution to the inequality $-2x \geq 5$?

A: To solve the inequality $-2x \geq 5$, we need to isolate the variable $x$ on one side of the inequality. We can do this by dividing both sides of the inequality by $-2$, which gives us $x \leq -\frac{5}{2}$.

Q: What is the solution to the inequality $x - 3 \geq 2$?

A: To solve the inequality $x - 3 \geq 2$, we need to isolate the variable $x$ on one side of the inequality. We can do this by adding $3$ to both sides of the inequality, which gives us $x \geq 5$.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities like $-3q \geq \frac{3}{4}$. We covered topics such as isolating the variable, simplifying the right-hand side, and reversing the direction of the inequality sign when multiplying or dividing by a negative number. We also provided examples of how to solve linear inequalities and equations.

Tips and Tricks

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

• Always read the inequality carefully and understand what it is saying.
• Use inverse operations to isolate the variable on one side of the inequality.
• Be careful when multiplying or dividing both sides of an inequality by a negative number.
• Simplify the right-hand side of the inequality by dividing the numerator and denominator by their greatest common divisor.

Common Mistakes

Here are some common mistakes to avoid when solving inequalities like this one:

• Not reading the inequality carefully and understanding what it is saying.
• Not using inverse operations to isolate the variable on one side of the inequality.
• Not being careful when multiplying or dividing both sides of an inequality by a negative number.
• Not simplifying the right-hand side of the inequality by dividing the numerator and denominator by their greatest common divisor.

Final Answer

The final answer to the inequality $-3q \geq \frac{3}{4}$ is $q \leq \frac{1}{4}$.