Solve The Inequality: $\[ \frac{2}{3} X - \frac{1}{5} \ \textgreater \ 1 \\]Find \[$x \ \textgreater \ \$\]

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Introduction

Inequalities are mathematical expressions that compare two values, often using greater than or less than symbols. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality $\frac{2}{3} x - \frac{1}{5} \ \textgreater \ 1$.

Understanding the Inequality

The given inequality is $\frac{2}{3} x - \frac{1}{5} \ \textgreater \ 1$. 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.

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

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

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

Simplifying the left-hand side, we get:

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

Isolating the Variable

Now that we have isolated the term with the variable, we can proceed to isolate the variable $x$ on one side of the inequality sign. To do this, we need to multiply both sides of the inequality by the reciprocal of $\frac{2}{3}$, which is $\frac{3}{2}$.

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

When we multiply both sides of the inequality by $\frac{3}{2}$, we get:

$\frac{3}{2} \cdot \frac{2}{3} x \ \textgreater \ \frac{3}{2} \cdot \frac{6}{5}$

Simplifying the left-hand side, we get:

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

Simplifying the Right-Hand Side

The right-hand side of the inequality is $\frac{18}{10}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

Simplifying the Fraction

When we divide both the numerator and the denominator by 2, we get:

$\frac{18}{10} = \frac{9}{5}$

So, the inequality becomes:

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

Conclusion

In this article, we solved the inequality $\frac{2}{3} x - \frac{1}{5} \ \textgreater \ 1$ by isolating the variable $x$ on one side of the inequality sign. We added $\frac{1}{5}$ to both sides of the inequality and then multiplied both sides by the reciprocal of $\frac{2}{3}$ to isolate the variable. Finally, we simplified the right-hand side of the inequality by dividing both the numerator and the denominator by their greatest common divisor. The solution to the inequality is $x \ \textgreater \ \frac{9}{5}$.

Frequently Asked Questions

• What is the solution to the inequality $\frac{2}{3} x - \frac{1}{5} \ \textgreater \ 1$?
  • The solution to the inequality is $x \ \textgreater \ \frac{9}{5}$.
• How do I solve an inequality?
  • To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
• What is the difference between an inequality and an equation?
  • An inequality is a mathematical expression that compares two values, often using greater than or less than symbols. An equation is a mathematical expression that states that two values are equal.

Step-by-Step Guide to Solving Inequalities

1. Read the inequality carefully: Make sure you understand what the inequality is saying.
2. Isolate the variable: Add or subtract the same value from both sides of the inequality, or multiply or divide both sides of the inequality by the same non-zero value.
3. Simplify the inequality: Simplify the right-hand side of the inequality by dividing both the numerator and the denominator by their greatest common divisor.
4. Write the solution: Write the solution to the inequality in the form $x \ \textgreater \ \text{value}$ or $x \ \textless \ \text{value}$.

Common Inequality Symbols

• $\ \textgreater \ $: greater than
• $\ \textless \ $: less than
• $\ \textgreater \ = \ $: greater than or equal to
• $\ \textless \ = \ $: less than or equal to

Conclusion

Solving inequalities involves isolating the variable on one side of the inequality sign. By following the steps outlined in this article, you can solve inequalities and write the solution in the form $x \ \textgreater \ \text{value}$ or $x \ \textless \ \text{value}$. Remember to read the inequality carefully, isolate the variable, simplify the inequality, and write the solution. With practice, you will become proficient in solving inequalities and will be able to apply this skill to a wide range of mathematical problems.

Introduction

In our previous article, we discussed how to solve the inequality $\frac{2}{3} x - \frac{1}{5} \ \textgreater \ 1$. In this article, we will provide a Q&A guide to help you understand the concepts and steps involved in solving inequalities.

Q&A Guide

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

A: An inequality is a mathematical expression that compares two values, often using greater than or less than symbols. An equation is a mathematical expression that states that two values are equal.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the solution to the inequality $\frac{2}{3} x - \frac{1}{5} \ \textgreater \ 1$?

A: The solution to the inequality is $x \ \textgreater \ \frac{9}{5}$.

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

A: To simplify the right-hand side of an inequality, you need to divide both the numerator and the denominator by their greatest common divisor.

Q: What are some common inequality symbols?

A: Some common inequality symbols include:

• $\ \textgreater \ $: greater than
• $\ \textless \ $: less than
• $\ \textgreater \ = \ $: greater than or equal to
• $\ \textless \ = \ $: less than or equal to

Q: How do I write the solution to an inequality?

A: To write the solution to an inequality, you need to write the solution in the form $x \ \textgreater \ \text{value}$ or $x \ \textless \ \text{value}$.

Q: What is the step-by-step guide to solving inequalities?

A: The step-by-step guide to solving inequalities is as follows:

1. Read the inequality carefully: Make sure you understand what the inequality is saying.
2. Isolate the variable: Add or subtract the same value from both sides of the inequality, or multiply or divide both sides of the inequality by the same non-zero value.
3. Simplify the inequality: Simplify the right-hand side of the inequality by dividing both the numerator and the denominator by their greatest common divisor.
4. Write the solution: Write the solution to the inequality in the form $x \ \textgreater \ \text{value}$ or $x \ \textless \ \text{value}$.

Frequently Asked Questions

• What is the solution to the inequality $\frac{2}{3} x - \frac{1}{5} \ \textgreater \ 1$?
  • The solution to the inequality is $x \ \textgreater \ \frac{9}{5}$.
• How do I solve an inequality?
  • To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.
• What is the difference between an inequality and an equation?
  • An inequality is a mathematical expression that compares two values, often using greater than or less than symbols. An equation is a mathematical expression that states that two values are equal.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and steps involved in solving inequalities. We discussed how to solve the inequality $\frac{2}{3} x - \frac{1}{5} \ \textgreater \ 1$ and provided a step-by-step guide to solving inequalities. We also answered some frequently asked questions about inequalities. With this guide, you should be able to solve inequalities and write the solution in the form $x \ \textgreater \ \text{value}$ or $x \ \textless \ \text{value}$.