Solve $10p - 8 \geq 12$.

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 a specific linear inequality, $10p - 8 \geq 12$, and provide a step-by-step guide on how to approach it.

Understanding Linear Inequalities

A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form $ax + b$, where $a$ and $b$ are constants, and $x$ is the variable. Linear inequalities can be written in the form $ax + b \geq c$, where $c$ is a constant.

The Given Inequality

The given inequality is $10p - 8 \geq 12$. This is a linear inequality, and our goal is to solve for $p$.

Step 1: Add 8 to Both Sides

To solve the inequality, we need to isolate the variable $p$. The first step is to add 8 to both sides of the inequality. This will eliminate the negative term on the left-hand side.

10p - 8 + 8 \geq 12 + 8

Simplifying the inequality, we get:

10p \geq 20

Step 2: Divide Both Sides by 10

Now that we have isolated the term with the variable, we need to divide both sides of the inequality by 10. This will give us the value of $p$.

\frac{10p}{10} \geq \frac{20}{10}

Simplifying the inequality, we get:

p \geq 2

Conclusion

In this article, we solved the linear inequality $10p - 8 \geq 12$ using a step-by-step approach. We added 8 to both sides of the inequality to eliminate the negative term, and then divided both sides by 10 to isolate the variable $p$. The final solution is $p \geq 2$.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following these steps and tips, you'll be able to solve linear inequalities with ease.

Common Mistakes to Avoid

When solving linear inequalities, it's easy to make mistakes. Here are some common mistakes to avoid:

1. Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving linear inequalities.
2. Not isolating the variable: Make sure to isolate the variable on one side of the inequality.
3. Not checking the direction of the inequality: Make sure to check the direction of the inequality after solving the inequality.

By avoiding these common mistakes, you'll be able to solve linear inequalities with confidence.

Real-World Applications

Linear inequalities have many real-world applications. Here are a few examples:

1. Finance: Linear inequalities can be used to model financial situations, such as determining the minimum amount of money needed to invest in a stock.
2. Science: Linear inequalities can be used to model scientific situations, such as determining the minimum amount of time needed to complete an experiment.
3. Engineering: Linear inequalities can be used to model engineering situations, such as determining the minimum amount of material needed to build a structure.

By understanding linear inequalities and how to solve them, you'll be able to apply this knowledge to real-world situations.

Conclusion

Introduction

In our previous article, we discussed how to solve linear inequalities using a step-by-step approach. In this article, we will provide a Q&A guide to help you better understand the concept of linear inequalities and how to solve them.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form $ax + b$, where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I know if an inequality is linear or not?

A: To determine if an inequality is linear or not, look for the following characteristics:

• The inequality must involve a linear expression, which is an expression that can be written in the form $ax + b$.
• The inequality must not involve any exponential or trigonometric functions.
• The inequality must not involve any absolute value or modulus functions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an expression or inequality. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I isolate the variable in a linear inequality?

A: To isolate the variable in a linear inequality, follow these steps:

1. Add or subtract the same value to both sides of the inequality to eliminate any constants.
2. Multiply or divide both sides of the inequality by the same value to eliminate any coefficients.
3. Repeat steps 1 and 2 until the variable is isolated on one side of the inequality.

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

A: A linear equation is an equation that involves a linear expression, which is an expression that can be written in the form $ax + b$. A linear inequality, on the other hand, is an inequality that involves a linear expression.

Q: Can I use the same methods to solve linear inequalities as I would to solve linear equations?

A: While some methods may be similar, there are key differences between solving linear inequalities and solving linear equations. When solving linear inequalities, you must consider the direction of the inequality and ensure that the solution set is correct.

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

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

• Not following the order of operations (PEMDAS)
• Not isolating the variable
• Not checking the direction of the inequality
• Not considering the solution set

Q: How do I check the solution set of a linear inequality?

A: To check the solution set of a linear inequality, follow these steps:

1. Write the inequality in the form $ax + b \geq c$ or $ax + b \leq c$.
2. Determine the direction of the inequality.
3. Check if the solution set is correct by plugging in test values.

Conclusion

In conclusion, solving linear inequalities requires a clear understanding of the concept and the ability to apply the correct methods. By following the order of operations (PEMDAS) and isolating the variable, you can solve linear inequalities with ease. Remember to check the direction of the inequality and consider the solution set to ensure that your solution is correct.

Additional Resources

For more information on solving linear inequalities, check out the following resources:

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

Practice Problems

Try solving the following linear inequalities:

1. $2x + 3 \geq 5$
2. $x - 2 \leq 3$
3. $4x + 2 \geq 10$

Answer Key

1. $x \geq 1$
2. $x \leq 5$
3. $x \geq 2$