Solve For $y$. $\frac{y}{9} \ \textgreater \ 2$

Feb 27, 2025 by ADMIN 52 views

$Solve for $y$: $\frac{y}{9} \ \textgreater \ 2$$

Introduction

When dealing with inequalities involving fractions, it's essential to isolate the variable to solve for its value. In this case, we have the inequality $\frac{y}{9} \ \textgreater \ 2$ , and our goal is to solve for $y$ . This involves manipulating the inequality to isolate $y$ on one side of the inequality sign.

Step 1: Multiply Both Sides by 9

To isolate $y$ , we need to get rid of the fraction. We can do this by multiplying both sides of the inequality by 9. This will cancel out the denominator on the left side, leaving us with just $y$ .

\frac{y}{9} \  \textgreater \  2
\implies y \  \textgreater \  18

Step 2: Simplify the Inequality

After multiplying both sides by 9, we get $y \ \textgreater \ 18$ . This is the simplified form of the inequality.

Step 3: Write the Solution in Interval Notation

The solution to the inequality can be written in interval notation as $(18, \infty)$ . This indicates that $y$ is greater than 18 and can take on any value in the interval $(18, \infty)$ .

Conclusion

In conclusion, to solve the inequality $\frac{y}{9} \ \textgreater \ 2$ , we multiplied both sides by 9 to isolate $y$ . This resulted in the simplified inequality $y \ \textgreater \ 18$ , which can be written in interval notation as $(18, \infty)$ .

Examples and Applications

Here are a few examples and applications of solving inequalities involving fractions:

Example 1: Solve the inequality $\frac{x}{4} \ \textless \ 3$ $4x \textless 3$ .
- Multiply both sides by 4: $x \ \textless \ 12$
- Write the solution in interval notation: $(-\infty, 12)$
Example 2: Solve the inequality $\frac{y}{6} \ \textgreater \ 4$ $6y \textgreater 4$ .
- Multiply both sides by 6: $y \ \textgreater \ 24$
- Write the solution in interval notation: $(24, \infty)$

Tips and Tricks

Here are a few tips and tricks for solving inequalities involving fractions:

Tip 1: When multiplying both sides of an inequality by a negative number, flip the direction of the inequality sign.
Tip 2: When dividing both sides of an inequality by a negative number, flip the direction of the inequality sign.
Tip 3: When multiplying both sides of an inequality by a fraction, multiply the numerator and denominator by the same number.

Common Mistakes to Avoid

Here are a few common mistakes to avoid when solving inequalities involving fractions:

Mistake 1: Failing to flip the direction of the inequality sign when multiplying or dividing both sides by a negative number.
Mistake 2: Failing to multiply both sides of the inequality by the same number when multiplying both sides by a fraction.
Mistake 3: Failing to write the solution in interval notation.

Real-World Applications

Solving inequalities involving fractions has many real-world applications, including:

Finance: In finance, inequalities involving fractions are used to calculate interest rates and investment returns.
Science: In science, inequalities involving fractions are used to model population growth and decay.
Engineering: In engineering, inequalities involving fractions are used to design and optimize systems.

Conclusion

In conclusion, solving inequalities involving fractions is a crucial skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve inequalities involving fractions with ease. Remember to multiply both sides by the same number, flip the direction of the inequality sign when necessary, and write the solution in interval notation. With practice and patience, you'll become a pro at solving inequalities involving fractions in no time!

Introduction

In our previous article, we discussed how to solve the inequality $\frac{y}{9} \ \textgreater \ 2$ . We learned how to isolate $y$ by multiplying both sides of the inequality by 9. In this article, we'll answer some frequently asked questions about solving inequalities involving fractions.

Q&A

Q: What is the first step in solving an inequality involving a fraction?

A: The first step in solving an inequality involving a fraction is to multiply both sides of the inequality by the denominator of the fraction. This will cancel out the fraction and leave us with a simpler inequality.

Q: What happens when we multiply both sides of an inequality by a negative number?

A: When we multiply both sides of an inequality by a negative number, we need to flip the direction of the inequality sign. For example, if we have the inequality $x \ \textless \ 5$ and we multiply both sides by -1, we get $-x \ \textgreater \ -5$ .

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

A: To write the solution to an inequality in interval notation, we need to determine the values of the variable that satisfy the inequality. For example, if we have the inequality $x \ \textgreater \ 2$ , we can write the solution in interval notation as $(2, \infty)$ .

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses a strict inequality symbol, such as $>$ or $<$ . A non-strict inequality is an inequality that uses a non-strict inequality symbol, such as $\geq$ or $\leq$ . For example, the inequality $x \ \textgreater \ 2$ is a strict inequality, while the inequality $x \ \geq \ 2$ is a non-strict inequality.

Q: How do we solve an inequality involving a fraction with a negative denominator?

A: To solve an inequality involving a fraction with a negative denominator, we need to multiply both sides of the inequality by the negative denominator. This will cancel out the fraction and leave us with a simpler inequality. We also need to flip the direction of the inequality sign.

Q: What is the importance of solving inequalities involving fractions?

A: Solving inequalities involving fractions is an essential skill in mathematics and has many real-world applications. It is used in finance, science, and engineering to model and analyze complex systems.