Solve The Inequality For { Y $} : : : { 4y + 3 \ \textless \ 2y + 11 \}

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. A linear inequality is an inequality that can be written in the form of a linear equation, but with an inequality symbol instead of an equal sign. In this article, we will focus on solving linear inequalities of the form $ax + b < cx + d$, where $a$, $b$, $c$, and $d$ are constants. We will use the given inequality $4y + 3 < 2y + 11$ as an example to demonstrate the steps involved in solving linear inequalities.

Understanding the Inequality

Before we start solving the inequality, let's understand what it means. The given inequality is $4y + 3 < 2y + 11$. This means that the expression $4y + 3$ is less than the expression $2y + 11$. To solve this inequality, we need to isolate the variable $y$ on one side of the inequality symbol.

Step 1: Subtract 2y from Both Sides

To isolate the variable $y$, we need to get rid of the term $2y$ on the right-hand side of the inequality. We can do this by subtracting $2y$ from both sides of the inequality. This gives us:

$$4y - 2y + 3 < 2y - 2y + 11$$

Simplifying the left-hand side, we get:

$$2y + 3 < 11$$

Step 2: Subtract 3 from Both Sides

Next, we need to get rid of the constant term $3$ on the left-hand side of the inequality. We can do this by subtracting $3$ from both sides of the inequality. This gives us:

$$2y + 3 - 3 < 11 - 3$$

Simplifying both sides, we get:

$$2y < 8$$

Step 3: Divide Both Sides by 2

Finally, we need to isolate the variable $y$ by dividing both sides of the inequality by $2$. This gives us:

$$\frac{2y}{2} < \frac{8}{2}$$

Simplifying both sides, we get:

$$y < 4$$

Conclusion

In this article, we solved the linear inequality $4y + 3 < 2y + 11$ by following the steps outlined above. We subtracted $2y$ from both sides, then subtracted $3$ from both sides, and finally divided both sides by $2$. The solution to the inequality is $y < 4$. This means that any value of $y$ that is less than $4$ satisfies the inequality.

Real-World Applications

Linear inequalities have numerous real-world applications in various fields such as economics, finance, and engineering. For example, in economics, linear inequalities can be used to model the relationship between the price of a commodity and its demand. In finance, linear inequalities can be used to determine the minimum amount of money required to invest in a particular stock. 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 steps outlined above. Here are some tips and tricks to keep in mind:

• Always subtract the same value from both sides of the inequality.
• Always divide both sides of the inequality by the same value.
• Be careful when multiplying or dividing both sides of the inequality by a negative value, as this can change the direction of the inequality symbol.
• Use a calculator to check your solutions and ensure that they satisfy the original inequality.

Common Mistakes

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

• Subtracting the wrong value from both sides of the inequality.
• Dividing both sides of the inequality by a value that is not a factor of the left-hand side.
• Changing the direction of the inequality symbol when multiplying or dividing both sides by a negative value.
• Not checking the solutions to ensure that they satisfy the original inequality.

Conclusion

In conclusion, solving linear inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined above and avoiding common mistakes, you can solve linear inequalities with confidence. Remember to always check your solutions and ensure that they satisfy the original inequality. With practice and patience, you can become proficient in solving linear inequalities and apply them to real-world problems.