Solve For T T T .1. $3t - 9 \ \textless \ 6$2. − 3 T \textless 12 -3t \ \textless \ 12 − 3 T \textless 12 Possible Solutions:A. − 4 \textless T \textless 5 -4 \ \textless \ T \ \textless \ 5 − 4 \textless T \textless 5 B. 5 \textless T \textless − 4 5 \ \textless \ T \ \textless \ -4 5 \textless T \textless − 4 C. $t \ \textless \

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving two linear inequalities and finding the possible solutions for the variable $t$. We will use a step-by-step approach to solve each inequality and provide a clear explanation of the solution process.

Solving the First Inequality

$3t - 9 < 6$

To solve this inequality, we need to isolate the variable $t$ on one side of the inequality sign. We can start by adding $9$ to both sides of the inequality:

$$3t - 9 + 9 < 6 + 9$$

This simplifies to:

$$3t < 15$$

Next, we can divide both sides of the inequality by $3$ to solve for $t$:

$$\frac{3t}{3} < \frac{15}{3}$$

This simplifies to:

$$t < 5$$

Therefore, the solution to the first inequality is $t < 5$.

Solving the Second Inequality

$-3t < 12$

To solve this inequality, we can start by dividing both sides of the inequality by $-3$. However, when we divide by a negative number, we need to reverse the direction of the inequality sign:

$$\frac{-3t}{-3} > \frac{12}{-3}$$

This simplifies to:

$$t > -4$$

Therefore, the solution to the second inequality is $t > -4$.

Finding the Possible Solutions

Now that we have solved both inequalities, we need to find the possible solutions for the variable $t$. We can do this by combining the solutions to both inequalities:

$$-4 < t < 5$$

Therefore, the possible solutions for the variable $t$ are $-4 < t < 5$.

Conclusion

Solving linear inequalities requires a step-by-step approach and a clear understanding of the solution process. By following the steps outlined in this article, we can solve linear inequalities and find the possible solutions for the variable $t$. In this case, we found that the possible solutions for the variable $t$ are $-4 < t < 5$.

Discussion

• What are some common mistakes students make when solving linear inequalities?
• How can students use technology to solve linear inequalities?
• What are some real-world applications of linear inequalities?

Answer Key

1. $3t - 9 < 6$
   • $t < 5$
2. $-3t < 12$
   • $t > -4$
3. Possible solutions
   • $-4 < t < 5$

Additional Resources

• Khan Academy: Linear Inequalities
• Mathway: Linear Inequalities
• IXL: Linear Inequalities

References

• Larson, R., & Hostetler, R. P. (2015). College algebra. Cengage Learning.
• Sullivan, M. (2016). College algebra. Pearson Education.

About the Author

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In our previous article, we provided a step-by-step guide on solving linear inequalities and finding the possible solutions for the variable $t$. In this article, we will answer some of the most frequently asked questions about solving linear inequalities.

Q&A

Q: What is the difference between a linear equation and a linear inequality?

A: A linear equation is an equation in which the highest power of the variable is 1, and the equation is true for a specific value of the variable. A linear inequality, on the other hand, is an inequality in which the highest power of the variable is 1, and the inequality is true for a range of values of the variable.

Q: How do I solve a linear inequality with a negative coefficient?

A: To solve a linear inequality with a negative coefficient, you need to reverse the direction of the inequality sign when you divide both sides of the inequality by the negative coefficient.

Q: What is the solution to the inequality $2x + 5 > 11$?

A: To solve this inequality, you need to isolate the variable $x$ on one side of the inequality sign. You can start by subtracting $5$ from both sides of the inequality:

$$2x + 5 - 5 > 11 - 5$$

This simplifies to:

$$2x > 6$$

Next, you can divide both sides of the inequality by $2$ to solve for $x$:

$$\frac{2x}{2} > \frac{6}{2}$$

This simplifies to:

$$x > 3$$

Therefore, the solution to the inequality is $x > 3$.

Q: How do I graph a linear inequality on a number line?

A: To graph a linear inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality. If the inequality is of the form $x > a$, you need to plot a point to the right of $a$. If the inequality is of the form $x < a$, you need to plot a point to the left of $a$.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality in which the inequality sign is strict, meaning that the solution is not equal to the value of the variable. A non-strict inequality, on the other hand, is an inequality in which the inequality sign is not strict, meaning that the solution is equal to the value of the variable.

Q: How do I solve a linear inequality with a fraction coefficient?

A: To solve a linear inequality with a fraction coefficient, you need to multiply both sides of the inequality by the reciprocal of the fraction coefficient to eliminate the fraction.

Q: What is the solution to the inequality $x - \frac{1}{2} < 3$?

A: To solve this inequality, you need to isolate the variable $x$ on one side of the inequality sign. You can start by adding $\frac{1}{2}$ to both sides of the inequality:

$$x - \frac{1}{2} + \frac{1}{2} < 3 + \frac{1}{2}$$

This simplifies to:

$$x < \frac{7}{2}$$

Therefore, the solution to the inequality is $x < \frac{7}{2}$.

Conclusion

Solving linear inequalities requires a clear understanding of the solution process and the ability to apply mathematical concepts to real-world problems. By following the steps outlined in this article, you can solve linear inequalities and find the possible solutions for the variable $t$. We hope that this Q&A guide has provided you with a better understanding of solving linear inequalities.

Discussion

• What are some common mistakes students make when solving linear inequalities?
• How can students use technology to solve linear inequalities?
• What are some real-world applications of linear inequalities?

Answer Key

1. $2x + 5 > 11$
   • $x > 3$
2. $x - \frac{1}{2} < 3$
   • $x < \frac{7}{2}$

Additional Resources

• Khan Academy: Linear Inequalities
• Mathway: Linear Inequalities
• IXL: Linear Inequalities

References

• Larson, R., & Hostetler, R. P. (2015). College algebra. Cengage Learning.
• Sullivan, M. (2016). College algebra. Pearson Education.

About the Author

[Your Name] is a mathematics educator with a passion for helping students understand complex mathematical concepts. With a background in mathematics and education, [Your Name] has developed a unique approach to teaching mathematics that emphasizes problem-solving and critical thinking.