Solve For $h$. H 7 − 2 \textless 1 \frac{h}{7} - 2 \ \textless \ 1 7 H ​ − 2 \textless 1

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving linear inequalities, specifically the inequality $\frac{h}{7} - 2 < 1$. We will break down the solution process into manageable steps, making it easy to understand and apply.

Understanding the Inequality

Before we dive into solving the inequality, let's understand what it means. The inequality $\frac{h}{7} - 2 < 1$ states that the expression $\frac{h}{7} - 2$ is less than 1. In other words, we want to find the values of $h$ that make the expression $\frac{h}{7} - 2$ less than 1.

Step 1: Add 2 to Both Sides

To solve the inequality, we need to isolate the variable $h$. The first step is to add 2 to both sides of the inequality. This will eliminate the constant term on the left-hand side.

$\frac{h}{7} - 2 + 2 < 1 + 2$

Simplifying the inequality, we get:

$\frac{h}{7} < 3$

Step 2: Multiply Both Sides by 7

Now that we have the inequality in the form $\frac{h}{7} < 3$, we need to eliminate the fraction. To do this, we can multiply both sides of the inequality by 7.

$\frac{h}{7} \cdot 7 < 3 \cdot 7$

Simplifying the inequality, we get:

$h < 21$

Step 3: Write the Solution in Interval Notation

The final step is to write the solution in interval notation. The inequality $h < 21$ means that $h$ can take on any value less than 21. In interval notation, this is written as $(-\infty, 21)$.

Conclusion

Solving linear inequalities requires a step-by-step approach. By adding 2 to both sides of the inequality, multiplying both sides by 7, and writing the solution in interval notation, we can find the values of $h$ that satisfy the inequality $\frac{h}{7} - 2 < 1$. This process can be applied to a wide range of linear inequalities, making it an essential skill for students and professionals in mathematics and related fields.

Real-World Applications

Linear inequalities have numerous real-world applications. For example, in finance, linear inequalities can be used to model investment portfolios and determine the optimal investment strategy. In engineering, linear inequalities can be used to design and optimize systems, such as electrical circuits and mechanical systems.

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 this order of operations, you can ensure that your solutions are accurate and reliable.

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: Failing to follow the order of operations can lead to incorrect solutions.
2. Not isolating the variable: Failing to isolate the variable can make it difficult to determine the solution.
3. Not checking the solution: Failing to check the solution can lead to incorrect conclusions.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Conclusion

Introduction

In our previous article, we explored the concept of solving linear inequalities, specifically the inequality $\frac{h}{7} - 2 < 1$. We broke down the solution process into manageable steps, making it easy to understand and apply. In this article, we will answer some frequently asked questions about solving linear inequalities.

Q: What is a linear inequality?

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

Q: How do I solve a linear inequality?

To solve a linear inequality, you need to follow these steps:

1. Add or subtract the same value to both sides of the inequality to isolate the variable.
2. Multiply or divide both sides of the inequality by the same non-zero value to eliminate the coefficient of the variable.
3. Write the solution in interval notation.

Q: What is interval notation?

Interval notation is a way of writing the solution to an inequality in a compact and concise form. For example, the solution to the inequality $x < 3$ can be written in interval notation as $(-\infty, 3)$.

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

To determine the solution to a linear inequality, you need to follow these steps:

1. Identify the variable and the constant term.
2. Add or subtract the same value to both sides of the inequality to isolate the variable.
3. Multiply or divide both sides of the inequality by the same non-zero value to eliminate the coefficient of the variable.
4. Write the solution in interval notation.

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

Some common mistakes to avoid when solving linear inequalities include:

1. Not following the order of operations.
2. Not isolating the variable.
3. Not checking the solution.

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

To check the solution to a linear inequality, you need to plug in a value from the solution set into the original inequality and verify that it is true.

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

Linear inequalities have numerous real-world applications, including:

1. Finance: Linear inequalities can be used to model investment portfolios and determine the optimal investment strategy.
2. Engineering: Linear inequalities can be used to design and optimize systems, such as electrical circuits and mechanical systems.
3. Science: Linear inequalities can be used to model physical systems and determine the optimal solution.

Q: How do I graph a linear inequality?

To graph a linear inequality, you need to follow these steps:

1. Graph the related linear equation.
2. Shade the region that satisfies the inequality.

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

A linear inequality is an inequality that can be written in the form $ax + b < c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. A linear equation is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Conclusion

Solving linear inequalities is a crucial skill for students and professionals in mathematics and related fields. By following the step-by-step approach outlined in this article, you can solve linear inequalities with confidence. Remember to follow the order of operations, isolate the variable, and check the solution to ensure accuracy and reliability. With practice and patience, you can become proficient in solving linear inequalities and apply them to real-world problems.