Solve The Inequality:${ -5x + 2y \ \textgreater \ -8 }$

Mar 13, 2025 by ADMIN 59 views

Solving the Inequality: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve them. 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 an inequality involves finding the values of the variable that make the inequality true. In this case, we will focus on solving the inequality $-5x + 2y > -8$ .

Understanding the Inequality

Before we start solving the inequality, let's understand what it means. The inequality $-5x + 2y > -8$ states that the expression $-5x + 2y$ is greater than $-8$ . This means that the value of $-5x + 2y$ can be any number greater than $-8$ . To solve this inequality, we need to isolate the variable $x$ and find the values of $x$ that make the inequality true.

Isolating the Variable

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

-5x + 2y - 2y > -8 - 2y

Simplifying the left-hand side, we get:

-5x > -8 - 2y

Solving for x

Now that we have isolated the variable $x$ , we can solve for $x$ . To do this, we need to get rid of the negative sign on the left-hand side of the inequality. We can do this by multiplying both sides of the inequality by $-1$ . This gives us:

5x < 8 + 2y

Dividing by 5

To solve for $x$ , we need to get rid of the coefficient $5$ on the left-hand side of the inequality. We can do this by dividing both sides of the inequality by $5$ . This gives us:

x < \frac{8 + 2y}{5}

Conclusion

In this article, we learned how to solve the inequality $-5x + 2y > -8$ . We started by understanding the inequality and isolating the variable $x$ . We then solved for $x$ by getting rid of the negative sign and dividing by the coefficient. The final solution is $x < \frac{8 + 2y}{5}$ . This means that the value of $x$ can be any number less than $\frac{8 + 2y}{5}$ .

Example

Let's consider an example to illustrate how to solve the inequality. Suppose we want to find the values of $x$ that satisfy the inequality $-5x + 2y > -8$ when $y = 3$ . We can substitute $y = 3$ into the inequality and solve for $x$ .

-5x + 2(3) > -8

Simplifying the left-hand side, we get:

-5x + 6 > -8

Subtracting $6$ from both sides, we get:

-5x > -14

Multiplying both sides by $-1$ , we get:

5x < 14

Dividing both sides by $5$ , we get:

x < \frac{14}{5}

Therefore, when $y = 3$ , the values of $x$ that satisfy the inequality are $x < \frac{14}{5}$ .

Graphing the Inequality

To visualize the solution to the inequality, we can graph the inequality on a coordinate plane. The inequality $-5x + 2y > -8$ can be graphed as a line with a slope of $-\frac{5}{2}$ and a y-intercept of $4$ . The solution to the inequality is the region above the line.

Applications of Inequalities

Inequalities have many real-world applications. For example, in economics, inequalities are used to model the relationship between supply and demand. In engineering, inequalities are used to design and optimize systems. In finance, inequalities are used to model the behavior of financial markets.

Conclusion

Final Thoughts

In conclusion, solving inequalities is an important skill in mathematics and has many real-world applications. By understanding how to solve inequalities, we can model and analyze complex systems and make informed decisions. We hope that this article has provided a clear and concise guide to solving inequalities and has inspired you to learn more about this important topic.

References

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Introduction

In our previous article, we learned how to solve the inequality $-5x + 2y > -8$ . We started by understanding the inequality and isolating the variable $x$ . We then solved for $x$ by getting rid of the negative sign and dividing by the coefficient. The final solution is $x < \frac{8 + 2y}{5}$ . This means that the value of $x$ can be any number less than $\frac{8 + 2y}{5}$ .

In this article, we will answer some frequently asked questions about solving inequalities. We will cover topics such as how to solve inequalities with multiple variables, how to graph inequalities, and how to apply inequalities to real-world problems.

Q&A

Q: How do I solve an inequality with multiple variables?

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

For example, consider the inequality $2x + 3y > 5$ . To solve for $x$ , we can subtract $3y$ from both sides of the inequality:

2x + 3y - 3y > 5 - 3y

Simplifying the left-hand side, we get:

2x > 5 - 3y

Dividing both sides by $2$ , we get:

x > \frac{5 - 3y}{2}

Q: How do I graph an inequality?

A: To graph an inequality, you need to find the boundary line and then determine which side of the line satisfies the inequality.

For example, consider the inequality $x - 2y > 3$ . To graph this inequality, we can find the boundary line by setting the inequality to an equation:

x - 2y = 3

This is a linear equation, and we can graph it as a line on a coordinate plane. The inequality $x - 2y > 3$ is satisfied by the region above the line.

Q: How do I apply inequalities to real-world problems?

A: Inequalities have many real-world applications. For example, in economics, inequalities are used to model the relationship between supply and demand. In engineering, inequalities are used to design and optimize systems. In finance, inequalities are used to model the behavior of financial markets.

For example, consider a company that wants to determine the maximum amount of money it can spend on advertising. The company has a budget of $100,000, and it wants to spend at least $50,000 on advertising. We can model this problem using an inequality:

x + y \leq 100,000

where $x$ is the amount spent on advertising and $y$ is the amount spent on other expenses. To find the maximum amount that can be spent on advertising, we can solve for $x$ :

x \leq 100,000 - y

Since the company wants to spend at least $50,000 on advertising, we can set up the inequality:

x \geq 50,000

Substituting this into the previous inequality, we get:

50,000 \leq 100,000 - y

Simplifying the inequality, we get:

y \leq 50,000

Therefore, the company can spend at most $50,000 on other expenses.

Conclusion

In this article, we answered some frequently asked questions about solving inequalities. We covered topics such as how to solve inequalities with multiple variables, how to graph inequalities, and how to apply inequalities to real-world problems. We hope that this article has provided a clear and concise guide to solving inequalities and has inspired you to learn more about this important topic.

Final Thoughts

In conclusion, solving inequalities is an important skill in mathematics and has many real-world applications. By understanding how to solve inequalities, we can model and analyze complex systems and make informed decisions. We hope that this article has provided a useful resource for anyone looking to learn more about solving inequalities.