Solve The Inequality For $t$:$0.1 - 2t \geq 0.7$A. $t \leq -0.3$ B. $t \geq -0.3$ C. $t \geq -0.4$ D. $t \leq -0.4$

Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality $0.1 - 2t \geq 0.7$ for the variable $t$.

Understanding the Inequality

The given inequality is $0.1 - 2t \geq 0.7$. To solve this inequality, we need to isolate the variable $t$ on one side of the inequality sign. The first step is to add $2t$ to both sides of the inequality to get rid of the negative term.

Step 1: Add $2t$ to Both Sides

0.1 - 2t ≥ 0.7
0.1 + 2t ≥ 0.7 + 2t
2t ≥ 0.6

Step 2: Divide Both Sides by 2

To isolate the variable $t$, we need to divide both sides of the inequality by 2.

2t ≥ 0.6
t ≥ 0.6 / 2
t ≥ 0.3

However, we need to be careful when dividing both sides of an inequality by a negative number. In this case, we are dividing by 2, which is a positive number. Therefore, the direction of the inequality remains the same.

Step 3: Write the Solution in Interval Notation

The solution to the inequality $0.1 - 2t \geq 0.7$ is $t \geq -0.3$. This can be written in interval notation as $(-\infty, -0.3]$.

Conclusion

Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we solved the inequality $0.1 - 2t \geq 0.7$ for the variable $t$. We added $2t$ to both sides of the inequality to get rid of the negative term, and then divided both sides by 2 to isolate the variable $t$. The solution to the inequality is $t \geq -0.3$.

Answer Key

The correct answer is B. $t \geq -0.3$.

Tips and Tricks

• When solving inequalities, always check the direction of the inequality sign.
• When dividing both sides of an inequality by a negative number, the direction of the inequality sign will change.
• When writing the solution in interval notation, use parentheses to indicate the direction of the inequality sign.

Common Mistakes

• Failing to check the direction of the inequality sign.
• Failing to divide both sides of the inequality by a negative number.
• Writing the solution in interval notation incorrectly.

Real-World Applications

Solving inequalities has many real-world applications, including:

• Finance: Solving inequalities can help you determine the minimum or maximum value of an investment.
• Science: Solving inequalities can help you determine the minimum or maximum value of a physical quantity, such as temperature or pressure.
• Engineering: Solving inequalities can help you determine the minimum or maximum value of a design parameter, such as the length of a beam or the diameter of a pipe.

Conclusion

Introduction

In our previous article, we discussed how to solve inequalities using step-by-step examples. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values using greater than, less than, greater than or equal to, or less than or equal to.

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. This involves adding or subtracting the same value to both sides of the inequality, and then 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 $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are constants and $x$ is the variable. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants and $x$ is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. This involves adding or subtracting the same value to both sides of the inequality, and then multiplying or dividing both sides by the same non-zero value.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression, if possible, and then use the sign of the quadratic expression to determine the solution set.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values of the variable that satisfy the inequality.

Q: How do I write the solution set of an inequality in interval notation?

A: To write the solution set of an inequality in interval notation, you need to use parentheses to indicate the direction of the inequality sign.

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

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

• Failing to check the direction of the inequality sign.
• Failing to divide both sides of the inequality by a negative number.
• Writing the solution set in interval notation incorrectly.

Q: How do I apply inequalities in real-world situations?

A: Inequalities have many real-world applications, including finance, science, and engineering. For example, you can use inequalities to determine the minimum or maximum value of an investment, or to determine the minimum or maximum value of a physical quantity, such as temperature or pressure.

Conclusion

Solving inequalities is an important skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to check the direction of the inequality sign, divide both sides of the inequality by a negative number, and write the solution set in interval notation correctly.

Frequently Asked Questions

• 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 $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are constants and $x$ is the variable. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are constants and $x$ is the variable.
• Q: How do I solve a linear inequality? A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. This involves adding or subtracting the same value to both sides of the inequality, and then multiplying or dividing both sides by the same non-zero value.
• Q: How do I solve a quadratic inequality? A: To solve a quadratic inequality, you need to factor the quadratic expression, if possible, and then use the sign of the quadratic expression to determine the solution set.

Additional Resources

• Inequality Solver: A online tool that can help you solve inequalities.
• Inequality Calculator: A online tool that can help you calculate the solution set of an inequality.
• Inequality Tutorial: A online tutorial that can help you learn how to solve inequalities.