Choosing the Correct Inequality

Oct 13, 2025 by talynt.org 129 views

Introduction

In this article, we will explore the concept of inequalities and how they can be used to represent relationships between numbers. We will focus on a specific problem that involves finding all the numbers that satisfy a given condition. The problem states that three times two less than a number is greater than or equal to five times the number. We will start by choosing the correct inequality that represents this relationship and then proceed to solve for the numbers that satisfy the condition.

Choosing the Correct Inequality

The problem states that three times two less than a number is greater than or equal to five times the number. Let's represent the number as $n$ . We can write the given relationship as an inequality:

3(2-n) \geq 5n

However, this is not the only possible inequality that represents the given relationship. We can also write it as:

6-3n \geq 5n

Or:

6 \geq 8n

Or:

\frac{6}{8} \geq n

Or:

\frac{3}{4} \geq n

Solving the Inequality

Now that we have chosen the correct inequality, we can proceed to solve for the numbers that satisfy the condition. We will use the inequality $6-3n \geq 5n$ .

First, we can add $3n$ to both sides of the inequality to get:

6 \geq 8n

Next, we can divide both sides of the inequality by $8$ to get:

\frac{6}{8} \geq n

Simplifying the fraction, we get:

\frac{3}{4} \geq n

Finding the Numbers that Satisfy the Condition

Now that we have the inequality $\frac{3}{4} \geq n$ , we can find the numbers that satisfy the condition. Since the inequality is greater than or equal to, we need to find all the numbers that are less than or equal to $\frac{3}{4}$ .

The numbers that satisfy the condition are all the numbers that are less than or equal to $\frac{3}{4}$ . This includes all the numbers in the interval $[0, \frac{3}{4}]$ .

Conclusion

In this article, we explored the concept of inequalities and how they can be used to represent relationships between numbers. We focused on a specific problem that involved finding all the numbers that satisfy a given condition. We chose the correct inequality that represents the given relationship and then proceeded to solve for the numbers that satisfy the condition. We found that the numbers that satisfy the condition are all the numbers that are less than or equal to $\frac{3}{4}$ .

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Choose the correct inequality that represents the given relationship.
Add $3n$ to both sides of the inequality to get $6 \geq 8n$ .
Divide both sides of the inequality by $8$ to get $\frac{6}{8} \geq n$ .
Simplify the fraction to get $\frac{3}{4} \geq n$ .
Find the numbers that satisfy the condition by finding all the numbers that are less than or equal to $\frac{3}{4}$ .

Example Solutions

Here are some example solutions to the problem:

If $n = 0$ , then $3(2-n) = 6$ and $5n = 0$ . Since $6 \geq 0$ , the inequality is satisfied.
If $n = \frac{3}{4}$ , then $3(2-n) = 3$ and $5n = \frac{15}{4}$ . Since $3 \geq \frac{15}{4}$ is not true, the inequality is not satisfied.
If $n = 1$ , then $3(2-n) = 3$ and $5n = 5$ . Since $3 \geq 5$ is not true, the inequality is not satisfied.

Graphical Representation

Here is a graphical representation of the solution:

The graph shows the inequality $\frac{3}{4} \geq n$ . The numbers that satisfy the condition are all the numbers that are less than or equal to $\frac{3}{4}$ .

Real-World Applications

The concept of inequalities and the solution to this problem have many real-world applications. For example:

In finance, inequalities can be used to model the relationship between the price of a stock and its expected return.
In engineering, inequalities can be used to model the relationship between the speed of a vehicle and its stopping distance.
In medicine, inequalities can be used to model the relationship between the dose of a medication and its effect on the body.