Solve The Inequality: ${ \frac{2x}{5} + 1 \ \textgreater \ \frac{x}{3} - 7 }$

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Introduction

Inequalities are mathematical expressions that compare two values, often with a greater than or less than symbol. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving the inequality $\frac{2x}{5} + 1 \ \textgreater \ \frac{x}{3} - 7$. We will break down the solution into manageable steps, using algebraic manipulations and logical reasoning to arrive at the final answer.

Understanding the Inequality

The given inequality is $\frac{2x}{5} + 1 \ \textgreater \ \frac{x}{3} - 7$. Our goal is to isolate the variable $x$ on one side of the inequality sign. To do this, we need to get rid of the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators, which is 15.

Step 1: Multiply Both Sides by 15

To eliminate the fractions, we multiply both sides of the inequality by 15. This gives us:

$$15 \left( \frac{2x}{5} + 1 \right) \ \textgreater \ 15 \left( \frac{x}{3} - 7 \right)$$

Using the distributive property, we can expand the left-hand side of the inequality:

$$\frac{15 \cdot 2x}{5} + 15 \cdot 1 \ \textgreater \ 15 \left( \frac{x}{3} - 7 \right)$$

Simplifying the left-hand side, we get:

$$6x + 15 \ \textgreater \ 15 \left( \frac{x}{3} - 7 \right)$$

Step 2: Distribute 15 on the Right-Hand Side

Now, we need to distribute 15 on the right-hand side of the inequality:

$$6x + 15 \ \textgreater \ 5x - 105$$

Step 3: Subtract 5x from Both Sides

To isolate the variable $x$ on one side of the inequality, we subtract 5x from both sides:

$$6x - 5x + 15 \ \textgreater \ 5x - 5x - 105$$

Simplifying the left-hand side, we get:

$$x + 15 \ \textgreater \ -105$$

Step 4: Subtract 15 from Both Sides

Now, we need to subtract 15 from both sides of the inequality:

$$x + 15 - 15 \ \textgreater \ -105 - 15$$

Simplifying the left-hand side, we get:

$$x \ \textgreater \ -120$$

Conclusion

In this article, we solved the inequality $\frac{2x}{5} + 1 \ \textgreater \ \frac{x}{3} - 7$ using algebraic manipulations and logical reasoning. We started by multiplying both sides of the inequality by 15 to eliminate the fractions, then distributed 15 on the right-hand side, subtracted 5x from both sides, and finally subtracted 15 from both sides to isolate the variable $x$ on one side of the inequality. The final solution is $x \ \textgreater \ -120$.

Tips and Tricks

When solving inequalities, it's essential to maintain the direction of the inequality sign. If you multiply or divide both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign.

Real-World Applications

Inequalities have numerous real-world applications in fields such as economics, finance, and engineering. For example, in economics, inequalities can be used to model the relationship between supply and demand, while in finance, inequalities can be used to calculate the interest rate on a loan. In engineering, inequalities can be used to design and optimize systems.

Common Mistakes

When solving inequalities, it's easy to make mistakes. Some common mistakes include:

• Not maintaining the direction of the inequality sign
• Not distributing the same value to both sides of the inequality
• Not simplifying the inequality correctly

To avoid these mistakes, it's essential to carefully read and understand the problem, then follow the steps outlined in this article.

Final Thoughts

Solving inequalities requires a deep understanding of algebraic manipulations and logical reasoning. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy. Remember to maintain the direction of the inequality sign, distribute the same value to both sides of the inequality, and simplify the inequality correctly. With practice and patience, you can become proficient in solving inequalities and apply them to real-world problems.

Introduction

In our previous article, we solved the inequality $\frac{2x}{5} + 1 \ \textgreater \ \frac{x}{3} - 7$ using algebraic manipulations and logical reasoning. In this article, we will answer some frequently asked questions about solving inequalities.

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

A: The first step in solving an inequality is to get rid of the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

Q: How do I know which direction to maintain when solving an inequality?

A: When solving an inequality, you need to maintain the direction of the inequality sign. If you multiply or divide both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign.

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

A: Solving an inequality involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality sign. Solving an equation involves isolating the variable on one side of the equation sign, with no direction.

Q: Can I use the same steps to solve a linear inequality as I would to solve a quadratic inequality?

A: No, the steps to solve a linear inequality are different from the steps to solve a quadratic inequality. Linear inequalities involve one variable, while quadratic inequalities involve two variables.

Q: How do I know if an inequality is linear or quadratic?

A: If an inequality involves one variable, it is a linear inequality. If an inequality involves two variables, it is a quadratic inequality.

Q: Can I use algebraic manipulations to solve an inequality?

A: Yes, algebraic manipulations can be used to solve an inequality. However, you need to be careful not to change the direction of the inequality sign.

Q: What is the final step in solving an inequality?

A: The final step in solving an inequality is to check your solution by plugging it back into the original inequality.

Q: Can I use a calculator to solve an inequality?

A: Yes, a calculator can be used to solve an inequality. However, you need to be careful not to change the direction of the inequality sign.

Q: How do I know if an inequality has a solution?

A: If an inequality has a solution, it means that there is a value of the variable that makes the inequality true. If an inequality does not have a solution, it means that there is no value of the variable that makes the inequality true.

Q: Can I use inequalities to model real-world problems?

A: Yes, inequalities can be used to model real-world problems. Inequalities can be used to describe relationships between variables, such as the relationship between supply and demand in economics.

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 the variable and check if the inequality is true or false.

Q: Can I use inequalities to solve systems of equations?

A: Yes, inequalities can be used to solve systems of equations. Inequalities can be used to describe relationships between variables, such as the relationship between two variables in a system of equations.

Q: How do I know if an inequality is linear or nonlinear?

A: If an inequality involves a linear function, it is a linear inequality. If an inequality involves a nonlinear function, it is a nonlinear inequality.

Q: Can I use inequalities to model optimization problems?

A: Yes, inequalities can be used to model optimization problems. Inequalities can be used to describe relationships between variables, such as the relationship between the cost and the profit in an optimization problem.

Q: How do I know if an inequality is a linear programming problem?

A: If an inequality involves a linear function and a set of constraints, it is a linear programming problem.

Q: Can I use inequalities to model game theory problems?

A: Yes, inequalities can be used to model game theory problems. Inequalities can be used to describe relationships between variables, such as the relationship between the payoffs in a game theory problem.

Q: How do I know if an inequality is a nonlinear programming problem?

A: If an inequality involves a nonlinear function and a set of constraints, it is a nonlinear programming problem.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We covered topics such as the first step in solving an inequality, maintaining the direction of the inequality sign, and using algebraic manipulations to solve an inequality. We also discussed how to use inequalities to model real-world problems, such as optimization problems and game theory problems. By following the steps outlined in this article, you can become proficient in solving inequalities and apply them to real-world problems.