Solve The Inequality: $ -5f \ \textgreater \ -20 $

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. In this article, we will focus on solving a linear inequality, specifically the inequality $ -5f \ \textgreater \ -20 $. Solving inequalities is an essential skill in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and more.

Understanding the Inequality

The given inequality is $ -5f \ \textgreater \ -20 $. To solve this inequality, we need to isolate the variable $f$ on one side of the inequality sign. The inequality sign $>$ indicates that the value of $f$ should be greater than the value obtained by dividing $-20$ by $-5$.

Isolating the Variable

To isolate the variable $f$, we need to perform the following steps:

1. Divide both sides of the inequality by $-5$.
2. Change the direction of the inequality sign, as dividing by a negative number changes the direction of the inequality.

Step 1: Divide Both Sides by $-5$

When we divide both sides of the inequality by $-5$, we get:

$$\frac{-5f}{-5} \ \textgreater \ \frac{-20}{-5}$$

Simplifying the left-hand side, we get:

$$f \ \textgreater \ 4$$

Step 2: Change the Direction of the Inequality Sign

Since we divided both sides of the inequality by a negative number, we need to change the direction of the inequality sign. Therefore, the correct inequality is:

$$f \ \textless \ 4$$

Conclusion

In conclusion, the solution to the inequality $ -5f \ \textgreater \ -20 $ is $f \ \textless \ 4$. This means that the value of $f$ should be less than $4$ to satisfy the given inequality.

Applications of Solving Inequalities

Solving inequalities has numerous applications in various fields. For example:

• In physics, inequalities are used to describe the motion of objects, such as the velocity of a particle or the acceleration of a body.
• In engineering, inequalities are used to design and optimize systems, such as electrical circuits or mechanical systems.
• In economics, inequalities are used to model the behavior of economic systems, such as the supply and demand of goods and services.

Tips for Solving Inequalities

Here are some tips for solving inequalities:

• Always read the inequality carefully and understand what it is asking.
• Isolate the variable on one side of the inequality sign.
• Change the direction of the inequality sign when dividing by a negative number.
• Check your solution by plugging in values to ensure that it satisfies the original inequality.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving inequalities:

• Not changing the direction of the inequality sign when dividing by a negative number.
• Not isolating the variable on one side of the inequality sign.
• Not checking the solution by plugging in values.

Real-World Examples

Here are some real-world examples of solving inequalities:

• A company produces a certain product, and the cost of production is $20 per unit. If the selling price is $30 per unit, how many units should the company produce to make a profit?
• A car travels from city A to city B at an average speed of 60 km/h. If the distance between the two cities is 240 km, how long will it take to travel from city A to city B?

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics that has numerous applications in various fields. By following the steps outlined in this article, you can solve linear inequalities and apply them to real-world problems. Remember to always read the inequality carefully, isolate the variable, and change the direction of the inequality sign when dividing by a negative number.

Introduction

In our previous article, we discussed how to solve linear inequalities, specifically the inequality $ -5f \ \textgreater \ -20 $. In this article, we will provide a Q&A section to help you better understand the concept of solving inequalities.

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two or more expressions using a comparison operator, such as $>$, $<$, $\geq$, or $\leq$.

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

A: An equation is a mathematical statement that states that two expressions are equal, whereas an inequality is a mathematical statement that compares two or more expressions using a comparison operator.

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. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What happens when I divide both sides of an inequality by a negative number?

A: When you divide both sides of an inequality by a negative number, you need to change the direction of the inequality sign. For example, if you have the inequality $x > 5$ and you divide both sides by $-2$, the resulting inequality would be $x < -\frac{5}{2}$.

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality. For example, if you have the inequality $x > 5$ and you add $3$ to both sides, the resulting inequality would be $x + 3 > 8$.

Q: Can I multiply or divide both sides of an inequality by the same non-zero value?

A: Yes, you can multiply or divide both sides of an inequality by the same non-zero value. For example, if you have the inequality $x > 5$ and you multiply both sides by $2$, the resulting inequality would be $2x > 10$.

Q: What is the solution to the inequality $x > 5$?

A: The solution to the inequality $x > 5$ is all real numbers greater than $5$. This can be represented as $x \in (5, \infty)$.

Q: What is the solution to the inequality $x < 5$?

A: The solution to the inequality $x < 5$ is all real numbers less than $5$. This can be represented as $x \in (-\infty, 5)$.

Q: Can I have multiple solutions to an inequality?

A: Yes, you can have multiple solutions to an inequality. For example, if you have the inequality $x > 5$ and $x < 10$, the solution would be all real numbers greater than $5$ and less than $10$. This can be represented as $x \in (5, 10)$.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality. If the inequality is of the form $x > a$, you would plot a point to the right of $a$. If the inequality is of the form $x < a$, you would plot a point to the left of $a$.

Q: Can I have a compound inequality?

A: Yes, you can have a compound inequality. A compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or". For example, the inequality $x > 5$ and $x < 10$ is a compound inequality.

Conclusion

In conclusion, solving inequalities is an essential skill in mathematics that has numerous applications in various fields. By following the steps outlined in this article, you can solve linear inequalities and apply them to real-world problems. Remember to always read the inequality carefully, isolate the variable, and change the direction of the inequality sign when dividing by a negative number.