Christina Works At A Bookstore And Earns $ 7.50 \$7.50 $7.50 Per Hour Plus A $ 2 \$2 $2 Bonus For Each Book She Sells. Christina Sold 15 Books. She Wants To Earn A Minimum Of $ 300 \$300 $300 .Which Inequality Represents This Situation, And What

Introduction

Christina works at a bookstore and earns a fixed hourly wage plus a bonus for each book she sells. In this scenario, we will explore the inequality that represents Christina's earnings situation. We will also determine the minimum number of books she needs to sell to earn at least $\$300$.

Christina's Earnings Situation

Christina earns $\$7.50$ per hour and a $\$2$ bonus for each book she sells. If she sells $x$ books, her total earnings can be represented as:

$$\text{Total Earnings} = 7.50 + 2x$$

However, this is not an inequality, as it represents a specific situation. To create an inequality, we need to consider the minimum earnings Christina wants to achieve, which is $\$300$. We can represent this as:

$$7.50 + 2x \geq 300$$

This inequality states that Christina's total earnings must be greater than or equal to $\$300$.

Solving the Inequality

To solve the inequality, we need to isolate the variable $x$. We can do this by subtracting $7.50$ from both sides of the inequality:

$$2x \geq 300 - 7.50$$

$$2x \geq 292.50$$

Next, we can divide both sides of the inequality by $2$:

$$x \geq \frac{292.50}{2}$$

$$x \geq 146.25$$

This means that Christina needs to sell at least $146.25$ books to earn at least $\$300$.

Christina's Actual Situation

However, Christina sold $15$ books, which is less than the minimum number of books she needs to sell to earn at least $\$300$. To determine her actual earnings, we can substitute $x = 15$ into the original equation:

$$\text{Total Earnings} = 7.50 + 2(15)$$

$$\text{Total Earnings} = 7.50 + 30$$

$$\text{Total Earnings} = 37.50$$

This means that Christina's actual earnings are $\$37.50$, which is less than her desired earnings of $\$300$.

Conclusion

In this scenario, we created an inequality to represent Christina's earnings situation. We solved the inequality to determine the minimum number of books she needs to sell to earn at least $\$300$. We also calculated Christina's actual earnings based on the number of books she sold. This example demonstrates how inequalities can be used to model real-world situations and solve problems.

Key Takeaways

• Inequalities can be used to model real-world situations and solve problems.
• To create an inequality, we need to consider the minimum or maximum value of a variable.
• We can solve inequalities by isolating the variable and using inverse operations.
• Inequalities can be used to determine the minimum or maximum value of a variable.

Mathematical Concepts

• Inequalities
• Linear equations
• Inverse operations
• Solving inequalities

Real-World Applications

• Modeling real-world situations
• Solving problems
• Determining minimum or maximum values
• Making informed decisions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Inequalities" by Michael Artin
  Christina's Bookstore Earnings Inequality: Q&A =====================================================

Introduction

In our previous article, we explored Christina's earnings situation at a bookstore and created an inequality to represent her earnings. We also solved the inequality to determine the minimum number of books she needs to sell to earn at least $\$300$. In this article, we will answer some frequently asked questions about Christina's earnings situation.

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

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 we create an inequality to represent a real-world situation?

A: To create an inequality, we need to identify the variable and the minimum or maximum value we want to achieve. We can then use mathematical operations to create an inequality that represents the situation.

Q: How do we solve an inequality?

A: To solve an inequality, we need to isolate the variable and use inverse operations to get the variable alone. We can then determine the minimum or maximum value of the variable.

Q: What is the minimum number of books Christina needs to sell to earn at least $\$300$?

A: According to the inequality we created, Christina needs to sell at least $146.25$ books to earn at least $\$300$.

Q: What is Christina's actual earnings if she sells $15$ books?

A: If Christina sells $15$ books, her actual earnings are $\$37.50$, which is less than her desired earnings of $\$300$.

Q: Can we use inequalities to model other real-world situations?

A: Yes, inequalities can be used to model many real-world situations, such as determining the minimum or maximum value of a variable, modeling population growth, or solving optimization problems.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

• Linear inequalities: These are inequalities that can be written in the form $ax + b \geq c$ or $ax + b \leq c$.
• Quadratic inequalities: These are inequalities that can be written in the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$.
• Rational inequalities: These are inequalities that can be written in the form $\frac{ax + b}{cx + d} \geq 0$ or $\frac{ax + b}{cx + d} \leq 0$.

Q: How do we graph inequalities on a number line?

A: To graph an inequality on a number line, we need to identify the minimum or maximum value of the variable and then plot a point on the number line. We can then use a test point to determine the direction of the inequality.

Conclusion

In this article, we answered some frequently asked questions about Christina's earnings situation at a bookstore. We also explored some common types of inequalities and how to graph them on a number line. We hope this article has been helpful in understanding inequalities and how to use them to model real-world situations.

Key Takeaways

• Inequalities can be used to model real-world situations and solve problems.
• To create an inequality, we need to identify the variable and the minimum or maximum value we want to achieve.
• We can solve inequalities by isolating the variable and using inverse operations.
• Inequalities can be used to determine the minimum or maximum value of a variable.

Mathematical Concepts

• Inequalities
• Linear equations
• Inverse operations
• Solving inequalities
• Graphing inequalities on a number line

Real-World Applications

• Modeling real-world situations
• Solving problems
• Determining minimum or maximum values
• Making informed decisions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Inequalities" by Michael Artin