Translate The Sentence Into An Inequality.A Number X X X Divided By 4 Is Less Than Or Equal To 16.

Introduction

In mathematics, inequalities are a fundamental concept used to describe relationships between numbers or expressions. They are often used to represent real-world situations, such as comparing the cost of two items or determining the maximum or minimum value of a quantity. In this article, we will focus on translating verbal descriptions into inequalities, specifically the given sentence: "A number $x$ divided by 4 is less than or equal to 16."

Understanding the Sentence

The given sentence can be broken down into three key components:

• A number $x$
• Divided by 4
• Less than or equal to 16

To translate this sentence into an inequality, we need to understand the relationship between these components.

Dividing a Number by 4

When we divide a number by 4, we are essentially finding the quotient of that number and 4. This operation can be represented mathematically as:

$$\frac{x}{4}$$

Less Than or Equal to 16

The phrase "less than or equal to" indicates that the result of the division (i.e., $\frac{x}{4}$) is either less than 16 or equal to 16. This can be represented mathematically as:

$$\frac{x}{4} \leq 16$$

Combining the Components

Now that we have broken down the sentence into its individual components, we can combine them to form the inequality:

$$\frac{x}{4} \leq 16$$

This inequality states that the number $x$ divided by 4 is less than or equal to 16.

Solving the Inequality

To solve the inequality, we need to isolate the variable $x$. We can do this by multiplying both sides of the inequality by 4:

$$\frac{x}{4} \leq 16$$

$$x \leq 64$$

This means that the value of $x$ is less than or equal to 64.

Example Applications

Inequalities like the one we have just solved have many real-world applications. Here are a few examples:

• Budgeting: Suppose you have a budget of $64 to spend on a trip. You want to know how much you can spend on a hotel room. If the hotel room costs $x$ dollars, you can use the inequality to determine the maximum amount you can spend.
• Time Management: Suppose you have 64 minutes to complete a task. You want to know how much time you have left to complete the task. If you have already spent $x$ minutes on the task, you can use the inequality to determine the maximum amount of time you have left.

Conclusion

In this article, we have learned how to translate a verbal description into an inequality. We have broken down the sentence into its individual components and combined them to form the inequality. We have also solved the inequality and provided example applications of its use. By understanding how to translate verbal descriptions into inequalities, we can better analyze and solve real-world problems.

Key Takeaways

• Inequalities are a fundamental concept in mathematics used to describe relationships between numbers or expressions.
• To translate a verbal description into an inequality, we need to break down the sentence into its individual components and combine them.
• Solving an inequality involves isolating the variable and determining the maximum or minimum value of the expression.

Further Reading

If you want to learn more about inequalities and how to solve them, here are some additional resources:

• Khan Academy: Inequalities
• Mathway: Inequality Solver
• Wolfram Alpha: Inequality Solver

Introduction

In our previous article, we explored how to translate verbal descriptions into inequalities. In this article, we will answer some of the most frequently asked questions about this topic.

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

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that says one expression is greater than, less than, or equal to another expression.

Q: How do I know which inequality symbol to use?

A: The inequality symbol you use depends on the relationship between the two expressions. If the expression on the left is greater than the expression on the right, use the greater-than symbol (>). If the expression on the left is less than the expression on the right, use the less-than symbol (<). If the expression on the left is equal to the expression on the right, use the equal-to symbol (=).

Q: Can I use inequalities to solve real-world problems?

A: Yes, inequalities are a powerful tool for solving real-world problems. They can be used to model situations such as budgeting, time management, and more.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable and determine the maximum or minimum value of the expression. This can be done by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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

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

• Not isolating the variable
• Not considering the direction of the inequality
• Not checking for extraneous solutions

Q: Can I use inequalities to solve systems of equations?

A: Yes, inequalities can be used to solve systems of equations. This is known as a linear programming problem.

Q: How do I graph an inequality?

A: To graph an inequality, you need to graph the related equation and then shade the region that satisfies the inequality.

Q: What are some real-world applications of inequalities?

A: Some real-world applications of inequalities include:

• Budgeting
• Time management
• Optimization problems
• Linear programming

Q: Can I use inequalities to solve quadratic equations?

A: Yes, inequalities can be used to solve quadratic equations. This is known as a quadratic inequality.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then determine the sign of the expression in each factor.

Conclusion

In this article, we have answered some of the most frequently asked questions about translating verbal descriptions into inequalities. We have covered topics such as the difference between an inequality and an equation, how to solve an inequality, and some common mistakes to avoid. By mastering the skills we have learned in this article, you will be able to tackle more complex inequalities and real-world problems with confidence.

Key Takeaways

• Inequalities are a fundamental concept in mathematics used to describe relationships between numbers or expressions.
• To solve an inequality, you need to isolate the variable and determine the maximum or minimum value of the expression.
• Inequalities can be used to solve real-world problems such as budgeting, time management, and optimization problems.

Further Reading

If you want to learn more about inequalities and how to solve them, here are some additional resources:

• Khan Academy: Inequalities
• Mathway: Inequality Solver
• Wolfram Alpha: Inequality Solver

By practicing and mastering the skills we have learned in this article, you will be able to tackle more complex inequalities and real-world problems with confidence.