Solve The Inequality:$ \frac{2}{3} X - \frac{1}{5} \ \textgreater \ 1 }$Find The Value Of { X $}$ { X \ \textgreater \ $ $

Introduction

Inequalities are mathematical expressions that compare two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign and finding the range of values that satisfy the inequality. In this article, we will focus on solving the inequality $\frac{2}{3} x - \frac{1}{5} > 1$ and finding the value of $x$.

Understanding the Inequality

The given inequality is $\frac{2}{3} x - \frac{1}{5} > 1$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to add $\frac{1}{5}$ to both sides of the inequality to get rid of the negative term.

Adding $\frac{1}{5}$ to Both Sides

When we add $\frac{1}{5}$ to both sides of the inequality, we get:

$\frac{2}{3} x - \frac{1}{5} + \frac{1}{5} > 1 + \frac{1}{5}$

Simplifying the right-hand side, we get:

$\frac{2}{3} x > \frac{6}{5}$

Multiplying Both Sides by $\frac{3}{2}$

To isolate the variable $x$, we need to multiply both sides of the inequality by $\frac{3}{2}$. This will cancel out the coefficient of $x$ on the left-hand side.

$\frac{2}{3} x \times \frac{3}{2} > \frac{6}{5} \times \frac{3}{2}$

Simplifying both sides, we get:

$x > \frac{18}{10}$

Simplifying the Right-Hand Side

We can simplify the right-hand side by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

$x > \frac{9}{5}$

Conclusion

In conclusion, the value of $x$ that satisfies the inequality $\frac{2}{3} x - \frac{1}{5} > 1$ is $x > \frac{9}{5}$. This means that any value of $x$ greater than $\frac{9}{5}$ will satisfy the inequality.

Tips and Tricks

• When solving inequalities, it's essential to keep the direction of the inequality sign the same.
• When adding or subtracting a term from both sides of an inequality, make sure to add or subtract the same term from both sides.
• When multiplying or dividing both sides of an inequality by a negative number, flip the direction of the inequality sign.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields, including:

• Finance: Inequality solving is used to calculate interest rates, investment returns, and loan payments.
• Engineering: Inequality solving is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Inequality solving is used to model and analyze economic systems, including supply and demand curves.

Common Mistakes to Avoid

• Flipping the direction of the inequality sign when multiplying or dividing both sides by a negative number.
• Not adding or subtracting the same term from both sides of an inequality.
• Not keeping the direction of the inequality sign the same when solving inequalities.

Conclusion

Solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities and find the value of $x$ that satisfies the inequality. Remember to keep the direction of the inequality sign the same, add or subtract the same term from both sides, and avoid common mistakes when solving inequalities.

Introduction

In our previous article, we discussed how to solve inequalities and find the value of $x$ that satisfies the inequality. In this article, we will answer some frequently asked questions about solving inequalities and provide additional tips and tricks to help you master this skill.

Q&A

Q: What is the difference between solving an equation and solving an inequality?

A: Solving an equation involves finding the value of the variable that makes the equation true, whereas solving an inequality involves finding the range of values that satisfy the inequality.

Q: How do I know which direction to flip the inequality sign when multiplying or dividing both sides by a negative number?

A: When multiplying or dividing both sides of an inequality by a negative number, you need to flip the direction of the inequality sign. For example, if you have the inequality $x > 2$ and you multiply both sides by $-1$, the inequality becomes $x < -2$.

Q: Can I add or subtract the same term from both sides of an inequality?

A: Yes, you can add or subtract the same term from both sides of an inequality. For example, if you have the inequality $x > 2$ and you add $3$ to both sides, the inequality becomes $x + 3 > 5$.

Q: How do I simplify the right-hand side of an inequality?

A: You can simplify the right-hand side of an inequality by dividing both the numerator and the denominator by their greatest common divisor. For example, if you have the inequality $x > \frac{18}{10}$, you can simplify the right-hand side by dividing both the numerator and the denominator by $2$, resulting in $x > \frac{9}{5}$.

Q: Can I multiply or divide both sides of an inequality by a fraction?

A: Yes, you can multiply or divide both sides of an inequality by a fraction. However, you need to make sure that the fraction is not equal to zero. For example, if you have the inequality $x > 2$ and you multiply both sides by $\frac{1}{2}$, the inequality becomes $x > 1$.

Q: How do I know if an inequality is true or false?

A: To determine if an inequality is true or false, you need to plug in a value of $x$ that satisfies the inequality and check if the inequality is true. For example, if you have the inequality $x > 2$ and you plug in $x = 3$, the inequality is true.

Tips and Tricks

• When solving inequalities, it's essential to keep the direction of the inequality sign the same.
• When adding or subtracting a term from both sides of an inequality, make sure to add or subtract the same term from both sides.
• When multiplying or dividing both sides of an inequality by a negative number, flip the direction of the inequality sign.
• When simplifying the right-hand side of an inequality, divide both the numerator and the denominator by their greatest common divisor.
• When multiplying or dividing both sides of an inequality by a fraction, make sure the fraction is not equal to zero.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields, including:

• Finance: Inequality solving is used to calculate interest rates, investment returns, and loan payments.
• Engineering: Inequality solving is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Inequality solving is used to model and analyze economic systems, including supply and demand curves.

Common Mistakes to Avoid

• Flipping the direction of the inequality sign when multiplying or dividing both sides by a negative number.
• Not adding or subtracting the same term from both sides of an inequality.
• Not keeping the direction of the inequality sign the same when solving inequalities.
• Not simplifying the right-hand side of an inequality.
• Not checking if an inequality is true or false.

Conclusion

Solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the tips and tricks outlined in this article, you can master the skill of solving inequalities and apply it to various fields. Remember to keep the direction of the inequality sign the same, add or subtract the same term from both sides, and avoid common mistakes when solving inequalities.