Simplify The Inequality: ${ 12 \geq 6p + 6p }$

Mar 10, 2025 by ADMIN 48 views

Simplify the Inequality: 12 ≥ 6p + 6p

Introduction

In mathematics, inequalities are a fundamental concept that helps us compare the magnitude of two or more quantities. In this article, we will focus on simplifying the given inequality, 12 ≥ 6p + 6p, and explore the various steps involved in solving it. We will also discuss the importance of inequalities in real-life applications and provide examples to illustrate their usage.

Understanding the Inequality

The given inequality is 12 ≥ 6p + 6p. To simplify this inequality, we need to combine like terms and isolate the variable p. The first step is to combine the two terms on the right-hand side of the inequality, which are both multiples of 6p.

Combining Like Terms

When we combine like terms, we add or subtract the coefficients of the same variables. In this case, we have two terms, 6p and 6p, which are both multiples of 6p. To combine them, we add their coefficients, which are both 6.

# Combining like terms
6p + 6p = 12p

So, the inequality becomes 12 ≥ 12p.

Isolating the Variable

Now that we have combined like terms, we need to isolate the variable p. To do this, we divide both sides of the inequality by 12.

Dividing Both Sides

When we divide both sides of an inequality by a positive number, the direction of the inequality remains the same. However, when we divide both sides by a negative number, the direction of the inequality is reversed.

In this case, we are dividing both sides by 12, which is a positive number. Therefore, the direction of the inequality remains the same.

# Dividing both sides
12 ≥ 12p
12/12 ≥ (12p)/12
1 ≥ p

So, the simplified inequality is 1 ≥ p.

Conclusion

In this article, we simplified the given inequality, 12 ≥ 6p + 6p, by combining like terms and isolating the variable p. We also discussed the importance of inequalities in real-life applications and provided examples to illustrate their usage.

Inequalities are a fundamental concept in mathematics that helps us compare the magnitude of two or more quantities. They are used in various fields, including economics, engineering, and computer science, to make decisions and solve problems.

Real-Life Applications of Inequalities

Inequalities have numerous real-life applications in various fields. Here are a few examples:

Example 1: Budgeting

Suppose you have a budget of $100 to spend on groceries and entertainment. You want to allocate $50 for groceries and $30 for entertainment. However, you also want to save some money for emergencies. In this case, you can use an inequality to represent the situation.

Let x be the amount of money you save for emergencies. Then, the inequality can be written as:

100 ≥ 50 + 30 + x

To simplify this inequality, you can combine like terms and isolate the variable x.

# Simplifying the inequality
100 ≥ 80 + x
20 ≥ x

So, the simplified inequality is 20 ≥ x.

Example 2: Time and Distance

Suppose you are driving a car at a speed of 60 km/h. You want to reach a destination that is 240 km away. However, you also want to arrive at the destination before 4 pm. In this case, you can use an inequality to represent the situation.

Let t be the time it takes to reach the destination. Then, the inequality can be written as:

t ≤ 240/60

To simplify this inequality, you can divide both sides by 60.

# Simplifying the inequality
t ≤ 4

So, the simplified inequality is t ≤ 4.

Importance of Inequalities

Inequalities have numerous real-life applications, including budgeting, time and distance, and optimization problems. They are also used in various mathematical concepts, such as linear programming and game theory.

Conclusion

Final Thoughts

We hope that this article has provided you with a better understanding of inequalities and their importance in mathematics and real-life applications.

Introduction

In our previous article, we discussed the concept of inequalities and how to simplify them. In this article, we will answer some frequently asked questions (FAQs) about inequalities. We will cover topics such as the definition of an inequality, how to solve inequalities, and the importance of inequalities in real-life applications.

Q1: What is an inequality?

A1: An inequality is a statement that compares two or more quantities, indicating that one quantity is greater than, less than, or equal to another quantity. Inequalities are used to represent relationships between variables and are a fundamental concept in mathematics.

Q2: How do I solve an inequality?

A2: 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 to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Q3: What is the difference between an inequality and an equation?

A3: An equation is a statement that says two quantities are equal, while an inequality is a statement that compares two or more quantities, indicating that one quantity is greater than, less than, or equal to another quantity.

Q4: How do I determine the direction of the inequality sign?

A4: When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign is reversed. For example, if you have the inequality 2x ≥ 6 and you multiply both sides by -1, the inequality becomes -2x ≤ -6.

Q5: Can I use the same methods to solve linear inequalities as I would to solve linear equations?

A5: Yes, you can use the same methods to solve linear inequalities as you would to solve linear equations. However, you need to be careful when multiplying or dividing both sides of an inequality by a negative number, as this can change the direction of the inequality sign.

Q6: How do I graph an inequality on a number line?

A6: 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 closed circle at the point a. If the inequality is of the form x > a, you would plot an open circle at the point a.

Q7: Can I use inequalities to solve optimization problems?

A7: Yes, you can use inequalities to solve optimization problems. Inequalities can be used to represent constraints on variables, and can be used to find the maximum or minimum value of a function subject to those constraints.

Q8: How do I use inequalities to solve systems of equations?

A8: To use inequalities to solve systems of equations, you need to first solve each equation separately to find the values of the variables. Then, you can use the inequalities to determine which of the solutions are valid.

Q9: Can I use inequalities to solve problems in real-life applications?

A9: Yes, you can use inequalities to solve problems in real-life applications. Inequalities can be used to represent relationships between variables, and can be used to make decisions and solve problems in fields such as economics, engineering, and computer science.

Q10: How do I determine the importance of inequalities in real-life applications?

A10: To determine the importance of inequalities in real-life applications, you need to consider how inequalities are used to represent relationships between variables, and how they are used to make decisions and solve problems. Inequalities are a fundamental concept in mathematics, and are used in a wide range of fields to solve problems and make decisions.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about inequalities. We covered topics such as the definition of an inequality, how to solve inequalities, and the importance of inequalities in real-life applications. We hope that this article has provided you with a better understanding of inequalities and their importance in mathematics and real-life applications.

Final Thoughts

In conclusion, inequalities are a powerful tool in mathematics that helps us compare the magnitude of two or more quantities. They are used in various fields, including economics, engineering, and computer science, to make decisions and solve problems. In this article, we answered some frequently asked questions (FAQs) about inequalities, and we hope that this article has provided you with a better understanding of inequalities and their importance in mathematics and real-life applications.