Solve The Inequality:$\[ 3p + 2 \geq -10 \\]

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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 $3p + 2 \geq -10$. We will break down the steps involved in solving this inequality and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is $3p + 2 \geq -10$. This means that the expression $3p + 2$ is greater than or equal to $-10$. To solve this inequality, we need to isolate the variable $p$ on one side of the inequality sign.

Step 1: Subtract 2 from Both Sides

To isolate the term with the variable $p$, we need to subtract 2 from both sides of the inequality. This will give us:

$$3p + 2 - 2 \geq -10 - 2$$

Simplifying the left-hand side, we get:

$$3p \geq -12$$

Step 2: Divide Both Sides by 3

To isolate the variable $p$, we need to divide both sides of the inequality by 3. This will give us:

$$\frac{3p}{3} \geq \frac{-12}{3}$$

Simplifying the left-hand side, we get:

$$p \geq -4$$

Step 3: Writing the Solution in Interval Notation

The solution to the inequality $3p + 2 \geq -10$ is $p \geq -4$. We can write this in interval notation as $[-4, \infty)$.

Conclusion

Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we solved the inequality $3p + 2 \geq -10$ by subtracting 2 from both sides and then dividing both sides by 3. The solution to this inequality is $p \geq -4$, which can be written in interval notation as $[-4, \infty)$.

Tips and Tricks

• When solving inequalities, it's essential to keep the direction of the inequality sign the same throughout the solution process.
• When subtracting or adding a value to both sides of an inequality, make sure to subtract or add the same value to both sides.
• When dividing both sides of an inequality by a value, make sure to divide both sides by the same value.

Real-World Applications

Solving inequalities has numerous real-world applications in fields such as economics, finance, and engineering. For example, in economics, inequalities can be used to model the relationship between supply and demand. In finance, inequalities can be used to model the relationship between interest rates and investment returns. In engineering, inequalities can be used to model the relationship between physical quantities such as distance, speed, and time.

Common Mistakes to Avoid

• When solving inequalities, it's essential to avoid making mistakes such as flipping the direction of the inequality sign or subtracting or adding the wrong value to both sides.
• When dividing both sides of an inequality by a value, make sure to divide both sides by the same value.
• When writing the solution in interval notation, make sure to include all the values that satisfy the inequality.

Final Thoughts

Solving inequalities is an essential skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy. Remember to keep the direction of the inequality sign the same throughout the solution process, and make sure to subtract or add the same value to both sides. With practice and patience, you can become proficient in solving inequalities and apply this skill to real-world problems.

Additional Resources

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Frequently Asked Questions

• Q: What is the solution to the inequality $3p + 2 \geq -10$? A: The solution to the inequality $3p + 2 \geq -10$ is $p \geq -4$.
• Q: How do I solve inequalities? A: To solve inequalities, you need to isolate the variable on one side of the inequality sign. You can do this by subtracting or adding a value to both sides, or by dividing both sides by a value.
• Q: What are some common mistakes to avoid when solving inequalities? A: Some common mistakes to avoid when solving inequalities include flipping the direction of the inequality sign, subtracting or adding the wrong value to both sides, and dividing both sides by the wrong value.
  

Introduction

Solving inequalities is an essential skill in mathematics that has numerous real-world applications. In our previous article, we provided a step-by-step guide to solving the inequality $3p + 2 \geq -10$. In this article, we will provide a Q&A guide to help you better understand how to solve inequalities.

Q&A Guide

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. You can do this by subtracting or adding a value to both sides, or by dividing both sides by a 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 $ax + b \geq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality sign. You can do this by subtracting or adding a value to both sides, or by dividing both sides by a value.

Q: How do I solve a quadratic inequality?

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

Q: What is the solution to the inequality $x^2 + 4x + 4 \geq 0$?

A: The solution to the inequality $x^2 + 4x + 4 \geq 0$ is $x \geq -2$.

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

A: To write the solution to an inequality in interval notation, you need to include all the values that satisfy the inequality. For example, if the solution to an inequality is $x \geq -2$, you can write it in interval notation as $[-2, \infty)$.

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

A: Some common mistakes to avoid when solving inequalities include flipping the direction of the inequality sign, subtracting or adding the wrong value to both sides, and dividing both sides by the wrong value.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you need to plug in a value from the solution set into the original inequality and make sure it is true.

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

A: Solving inequalities has numerous real-world applications in fields such as economics, finance, and engineering. For example, in economics, inequalities can be used to model the relationship between supply and demand. In finance, inequalities can be used to model the relationship between interest rates and investment returns. In engineering, inequalities can be used to model the relationship between physical quantities such as distance, speed, and time.

Conclusion

Solving inequalities is an essential skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy. Remember to keep the direction of the inequality sign the same throughout the solution process, and make sure to subtract or add the same value to both sides. With practice and patience, you can become proficient in solving inequalities and apply this skill to real-world problems.

Additional Resources

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Frequently Asked Questions

• Q: What is the solution to the inequality $x^2 + 4x + 4 \geq 0$? A: The solution to the inequality $x^2 + 4x + 4 \geq 0$ is $x \geq -2$.
• 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. You can do this by subtracting or adding a value to both sides, or by dividing both sides by a value.
• Q: What are some common mistakes to avoid when solving inequalities? A: Some common mistakes to avoid when solving inequalities include flipping the direction of the inequality sign, subtracting or adding the wrong value to both sides, and dividing both sides by the wrong value.

Final Thoughts

Solving inequalities is an essential skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy. Remember to keep the direction of the inequality sign the same throughout the solution process, and make sure to subtract or add the same value to both sides. With practice and patience, you can become proficient in solving inequalities and apply this skill to real-world problems.