Solve For $p$. $55 \leq 50 + P$

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 linear inequalities, specifically the inequality $55 \leq 50 + p$. We will break down the solution step by step, using clear and concise language to ensure that readers understand the concept.

What are Linear Inequalities?

Linear inequalities are mathematical statements that compare two expressions, with one expression being greater than, less than, greater than or equal to, or less than or equal to the other expression. In the case of the inequality $55 \leq 50 + p$, we are comparing the expression $55$ to the expression $50 + p$.

The Basics of Solving Linear Inequalities

To solve a linear inequality, we need to isolate the variable (in this case, $p$) on one side of the inequality. We can do this by performing the same operations on both sides of the inequality. For example, if we have the inequality $2x \leq 5$, we can divide both sides by $2$ to get $x \leq \frac{5}{2}$.

Solving the Inequality $55 \leq 50 + p$

Now that we have a basic understanding of linear inequalities, let's solve the inequality $55 \leq 50 + p$. To do this, we need to isolate the variable $p$ on one side of the inequality.

Step 1: Subtract 50 from Both Sides

The first step in solving the inequality is to subtract $50$ from both sides. This will give us:

$$55 - 50 \leq 50 + p - 50$$

Simplifying the left-hand side, we get:

$$5 \leq p$$

Step 2: Write the Solution in Interval Notation

Now that we have isolated the variable $p$ on one side of the inequality, we can write the solution in interval notation. The solution is all values of $p$ that are greater than or equal to $5$. In interval notation, this is written as:

$$p \geq 5$$

Step 3: Check the Solution

To check the solution, we can plug in a value of $p$ that satisfies the inequality and see if it is true. Let's try plugging in $p = 10$:

$$55 \leq 50 + 10$$

Simplifying the right-hand side, we get:

$$55 \leq 60$$

This is true, so we know that the solution $p \geq 5$ is correct.

Conclusion

Solving linear inequalities is a crucial skill for students to master. In this article, we solved the inequality $55 \leq 50 + p$ by isolating the variable $p$ on one side of the inequality. We used clear and concise language to explain each step of the solution, and we checked the solution to ensure that it was correct. By following these steps, students can solve linear inequalities with confidence.

Common Mistakes to Avoid

When solving linear inequalities, there are several common mistakes to avoid. Here are a few:

• Not isolating the variable: Make sure to isolate the variable on one side of the inequality.
• Not checking the solution: Always check the solution to ensure that it is correct.
• Not using interval notation: Use interval notation to write the solution.

Practice Problems

Here are a few practice problems to help you master the concept of solving linear inequalities:

1. Solve the inequality $30 \leq 20 + p$.
2. Solve the inequality $p \geq 10$.
3. Solve the inequality $2x \leq 5$.

Answer Key

Here are the answers to the practice problems:

1. $p \geq 10$
2. $p \geq 10$
3. $x \leq \frac{5}{2}$

Conclusion

Introduction

In our previous article, we discussed the basics of solving linear inequalities and solved the inequality $55 \leq 50 + p$. In this article, we will provide a Q&A guide to help you master the concept of solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is a mathematical statement that compares two expressions, with one expression being greater than, less than, greater than or equal to, or less than or equal to the other expression.

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. You can do this by performing the same operations on both sides of the inequality.

Q: What are the steps to solve a linear inequality?

A: The steps to solve a linear inequality are:

1. Isolate the variable: Isolate the variable on one side of the inequality.
2. Check the solution: Check the solution to ensure that it is correct.
3. Use interval notation: Use interval notation to write the solution.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to a linear inequality. It is used to indicate the range of values that satisfy the inequality.

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

A: To write the solution in interval notation, you need to determine the range of values that satisfy the inequality. For example, if the solution is $p \geq 5$, you would write it as $[5, \infty)$.

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

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

• Not isolating the variable: Make sure to isolate the variable on one side of the inequality.
• Not checking the solution: Always check the solution to ensure that it is correct.
• Not using interval notation: Use interval notation to write the solution.

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

A: To check the solution to a linear inequality, you need to plug in a value of the variable that satisfies the inequality and see if it is true.

Q: What are some examples of linear inequalities?

A: Some examples of linear inequalities include:

• $2x \leq 5$
• $p \geq 10$
• $30 \leq 20 + p$

Q: How do I solve a linear inequality with fractions?

A: To solve a linear inequality with fractions, you need to follow the same steps as solving a linear inequality with integers. However, you may need to multiply both sides of the inequality by a common denominator to eliminate the fractions.

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

A: Linear inequalities have many real-world applications, including:

• Finance: Linear inequalities are used to model financial situations, such as determining the minimum amount of money needed to invest in a stock.
• Science: Linear inequalities are used to model scientific situations, such as determining the maximum amount of a substance that can be present in a solution.
• Engineering: Linear inequalities are used to model engineering situations, such as determining the minimum amount of material needed to build a structure.

Conclusion

Solving linear inequalities is a crucial skill for students to master. By following the steps outlined in this article, students can solve linear inequalities with confidence. Remember to isolate the variable, check the solution, and use interval notation to write the solution. With practice, you will become proficient in solving linear inequalities and be able to apply this skill to a wide range of mathematical problems.