Write A Situation For The Inequality 15 X − 20 ≤ 130 15x - 20 \leq 130 15 X − 20 ≤ 130 And Solve It.

Introduction

Linear inequalities are a fundamental concept in mathematics, and they have numerous applications in various fields, including economics, finance, and science. In this article, we will explore a real-world situation that can be modeled using the inequality $15x - 20 \leq 130$. We will first analyze the situation, then solve the inequality, and finally provide a conclusion.

Situation Analysis

Suppose a company is producing a new product, and the cost of producing x units of the product is given by the equation $15x - 20$. The company wants to ensure that the total cost does not exceed $130. We can model this situation using the inequality $15x - 20 \leq 130$.

Solving the Inequality

To solve the inequality, we need to isolate the variable x. We can start by adding 20 to both sides of the inequality:

$$15x - 20 + 20 \leq 130 + 20$$

This simplifies to:

$$15x \leq 150$$

Next, we can divide both sides of the inequality by 15:

$$\frac{15x}{15} \leq \frac{150}{15}$$

This simplifies to:

$$x \leq 10$$

Conclusion

In this article, we analyzed a real-world situation that can be modeled using the inequality $15x - 20 \leq 130$. We solved the inequality by isolating the variable x and found that x is less than or equal to 10. This means that the company can produce up to 10 units of the product without exceeding the total cost of $130.

Real-World Applications

Linear inequalities have numerous applications in real-world situations. For example, in finance, a company may want to determine the maximum amount of money it can borrow without exceeding a certain debt-to-equity ratio. In science, a researcher may want to determine the maximum amount of a certain substance that can be added to a solution without exceeding a certain concentration.

Tips for Solving Linear Inequalities

When solving linear inequalities, it's essential to follow the same steps as when solving linear equations. However, there are some key differences to keep in mind:

• When adding or subtracting a constant to both sides of the inequality, make sure to change the direction of the inequality sign.
• When multiplying or dividing both sides of the inequality by a negative number, make sure to change the direction of the inequality sign.
• When multiplying or dividing both sides of the inequality by a positive number, the direction of the inequality sign remains the same.

Common Mistakes to Avoid

When solving linear inequalities, there are several common mistakes to avoid:

• Failing to change the direction of the inequality sign when adding or subtracting a constant to both sides of the inequality.
• Failing to change the direction of the inequality sign when multiplying or dividing both sides of the inequality by a negative number.
• Failing to isolate the variable x.

Practice Problems

Here are some practice problems to help you reinforce your understanding of linear inequalities:

1. Solve the inequality $2x + 5 \leq 17$.
2. Solve the inequality $x - 3 \geq 2$.
3. Solve the inequality $4x - 2 \leq 10$.

Conclusion

In this article, we analyzed a real-world situation that can be modeled using the inequality $15x - 20 \leq 130$. We solved the inequality by isolating the variable x and found that x is less than or equal to 10. We also discussed the importance of linear inequalities in real-world situations and provided tips for solving them. Finally, we provided some practice problems to help you reinforce your understanding of linear inequalities.

Introduction

Linear inequalities are a fundamental concept in mathematics, and they have numerous applications in various fields, including economics, finance, and science. In this article, we will answer some frequently asked questions about linear inequalities, including how to solve them, common mistakes to avoid, and real-world applications.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression, which is an expression that can be written in the form ax + b, where a and b are constants and x is the variable.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable x. You can do this by adding or subtracting a constant to both sides of the inequality, or by multiplying or dividing both sides of the inequality by a positive or negative number.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that involves a linear expression, and it is equal to a constant. A linear inequality, on the other hand, is an inequality that involves a linear expression, and it is not equal to a constant.

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

A: When multiplying or dividing both sides of the inequality by a negative number, you need to change the direction of the inequality sign. For example, if the inequality is x ≤ 5 and you multiply both sides by -1, the inequality becomes -x ≥ -5.

Q: What is the importance of linear inequalities in real-world situations?

A: Linear inequalities have numerous applications in real-world situations, including finance, economics, and science. For example, a company may want to determine the maximum amount of money it can borrow without exceeding a certain debt-to-equity ratio, or a researcher may want to determine the maximum amount of a certain substance that can be added to a solution without exceeding a certain concentration.

Q: What are some common mistakes to avoid when solving linear inequalities?

A: Some common mistakes to avoid when solving linear inequalities include:

• Failing to change the direction of the inequality sign when adding or subtracting a constant to both sides of the inequality.
• Failing to change the direction of the inequality sign when multiplying or dividing both sides of the inequality by a negative number.
• Failing to isolate the variable x.

Q: How do I determine the solution set of a linear inequality?

A: To determine the solution set of a linear inequality, you need to find all the values of x that satisfy the inequality. You can do this by graphing the inequality on a number line or by using a calculator to find the solution set.

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

A: A strict inequality is an inequality that is written with a strict inequality sign, such as < or >. A non-strict inequality, on the other hand, is an inequality that is written with a non-strict inequality sign, such as ≤ or ≥.

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you need to find the values of x that satisfy all the inequalities in the system. You can do this by graphing the inequalities on a number line or by using a calculator to find the solution set.

Conclusion

In this article, we answered some frequently asked questions about linear inequalities, including how to solve them, common mistakes to avoid, and real-world applications. We also discussed the importance of linear inequalities in real-world situations and provided some tips for solving them.