What Is The Solution To The Inequality 75 \textgreater X 15 75 \ \textgreater \ \frac{x}{15} 75 \textgreater 15 X ​ ?A. X \textless 1 , 125 X \ \textless \ 1,125 X \textless 1 , 125 B. X \textgreater 1 , 125 X \ \textgreater \ 1,125 X \textgreater 1 , 125 C. X \textless 5 X \ \textless \ 5 X \textless 5 D. X \textgreater 5 X \ \textgreater \ 5 X \textgreater 5

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Understanding the Inequality

The given inequality is $75 \ \textgreater \ \frac{x}{15}$. To solve this inequality, we need to isolate the variable $x$ and determine the range of values that satisfy the given condition. The inequality can be rewritten as $\frac{x}{15} \ \textless \ 75$.

Isolating the Variable

To isolate the variable $x$, we need to multiply both sides of the inequality by 15. This will give us $x \ \textless \ 1125$. However, we need to be careful when multiplying or dividing both sides of an inequality by a negative number, as it will reverse the direction of the inequality.

Analyzing the Solution

The solution to the inequality is $x \ \textless \ 1125$. This means that any value of $x$ that is less than 1125 will satisfy the given inequality. In other words, the solution set is all real numbers less than 1125.

Comparing with the Options

Let's compare the solution with the given options:

• A. $x \ \textless \ 1125$
• B. $x \ \textgreater \ 1125$
• C. $x \ \textless \ 5$
• D. $x \ \textgreater \ 5$

The correct solution is option A, $x \ \textless \ 1125$.

Conclusion

In conclusion, the solution to the inequality $75 \ \textgreater \ \frac{x}{15}$ is $x \ \textless \ 1125$. This means that any value of $x$ that is less than 1125 will satisfy the given inequality.

Frequently Asked Questions

Q: What is the solution to the inequality $75 \ \textgreater \ \frac{x}{15}$?

A: The solution to the inequality is $x \ \textless \ 1125$.

Q: Why is the solution $x \ \textless \ 1125$?

A: The solution is $x \ \textless \ 1125$ because when we multiply both sides of the inequality by 15, we get $x \ \textless \ 1125$.

Q: What is the range of values that satisfy the given inequality?

A: The range of values that satisfy the given inequality is all real numbers less than 1125.

Step-by-Step Solution

Step 1: Rewrite the inequality

The given inequality is $75 \ \textgreater \ \frac{x}{15}$. We can rewrite it as $\frac{x}{15} \ \textless \ 75$.

Step 2: Multiply both sides by 15

To isolate the variable $x$, we need to multiply both sides of the inequality by 15. This will give us $x \ \textless \ 1125$.

Step 3: Analyze the solution

The solution to the inequality is $x \ \textless \ 1125$. This means that any value of $x$ that is less than 1125 will satisfy the given inequality.

Common Mistakes

Mistake 1: Reversing the direction of the inequality

When multiplying or dividing both sides of an inequality by a negative number, we need to reverse the direction of the inequality. However, in this case, we are multiplying both sides by a positive number, so the direction of the inequality remains the same.

Mistake 2: Not isolating the variable

To solve the inequality, we need to isolate the variable $x$. If we don't isolate the variable, we won't be able to determine the range of values that satisfy the given inequality.

Real-World Applications

Example 1: Budgeting

Suppose we have a budget of $75 and we want to know how much we can spend on a particular item. If the item costs $x, then we can set up the inequality $75 \ \textgreater \ \frac{x}{15}$. Solving this inequality, we get $x \ \textless \ 1125$. This means that we can spend up to $1125 on the item.

Example 2: Time Management

Suppose we have a certain amount of time to complete a task and we want to know how much time we have left. If we have spent $x amount of time on the task, then we can set up the inequality $75 \ \textgreater \ \frac{x}{15}$. Solving this inequality, we get $x \ \textless \ 1125$. This means that we have spent less than 1125 units of time on the task.

Conclusion

In conclusion, the solution to the inequality $75 \ \textgreater \ \frac{x}{15}$ is $x \ \textless \ 1125$. This means that any value of $x$ that is less than 1125 will satisfy the given inequality. We can apply this solution to real-world problems such as budgeting and time management.

Understanding Inequalities

Inequalities are mathematical statements that compare two expressions and indicate whether one is greater than, less than, or equal to the other. Inequalities are used to solve a wide range of problems in mathematics, science, and engineering.

Q&A on Inequalities

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two expressions and indicates whether one is greater than, less than, or equal to the other.

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. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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

A: An equation is a mathematical statement that states that two expressions are equal. An inequality, on the other hand, states that two expressions are not equal, but one is greater than, less than, or equal to the other.

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

A: When multiplying or dividing both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign.

Q: What is the solution to the inequality $75 \ \textgreater \ \frac{x}{15}$?

A: The solution to the inequality is $x \ \textless \ 1125$. This means that any value of $x$ that is less than 1125 will satisfy the given inequality.

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

A: Inequalities can be applied to a wide range of real-world problems, such as budgeting, time management, and optimization. For example, if you have a budget of $75 and you want to know how much you can spend on a particular item, you can set up an inequality to solve for the maximum amount you can spend.

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

A: Some common mistakes to avoid when solving inequalities include reversing the direction of the inequality sign when multiplying or dividing both sides by a negative number, and not isolating the variable on one side of the inequality sign.

Q: How do I determine the range of values that satisfy an inequality?

A: To determine the range of values that satisfy an inequality, you need to isolate the variable on one side of the inequality sign and then determine the values that satisfy the inequality.

Q: What is the importance of inequalities in mathematics and science?

A: Inequalities are an essential part of mathematics and science, and are used to solve a wide range of problems in fields such as physics, engineering, and economics.

Real-World Applications of Inequalities

Budgeting

Inequalities can be used to solve budgeting problems. For example, if you have a budget of $75 and you want to know how much you can spend on a particular item, you can set up an inequality to solve for the maximum amount you can spend.

Time Management

Inequalities can be used to solve time management problems. For example, if you have a certain amount of time to complete a task and you want to know how much time you have left, you can set up an inequality to solve for the remaining time.

Optimization

Inequalities can be used to solve optimization problems. For example, if you want to maximize the profit of a business and you have a certain amount of resources available, you can set up an inequality to solve for the optimal amount of resources to use.

Conclusion

In conclusion, inequalities are an essential part of mathematics and science, and are used to solve a wide range of problems in fields such as physics, engineering, and economics. By understanding how to solve inequalities and apply them to real-world problems, you can develop a deeper understanding of the world around you and make more informed decisions.

Common Inequalities

Linear Inequalities

A linear inequality is an inequality that can be written in the form $ax \ \textless \ b$ or $ax \ \textgreater \ b$, where $a$ and $b$ are constants.

Quadratic Inequalities

A quadratic inequality is an inequality that can be written in the form $ax^2 \ \textless \ b$ or $ax^2 \ \textgreater \ b$, where $a$ and $b$ are constants.

Polynomial Inequalities

A polynomial inequality is an inequality that can be written in the form $ax^n \ \textless \ b$ or $ax^n \ \textgreater \ b$, where $a$ and $b$ are constants and $n$ is a positive integer.

Conclusion

In conclusion, inequalities are an essential part of mathematics and science, and are used to solve a wide range of problems in fields such as physics, engineering, and economics. By understanding how to solve inequalities and apply them to real-world problems, you can develop a deeper understanding of the world around you and make more informed decisions.