Question

You tune and restore pianos. As a piano tuner you charge \(\$ 75\) per tuning. The expenses for your piano restoration business are \(\$ 2600\) per month. Which of the following inequalities could you use to find the number of pianos \(p\) you must tune per month in order to at least meet your business expenses? (A) \(75 p \leq 2600\) (B) \(75 p \geq 2600\) (C) \(\frac{75}{p} \geq 2600\) (D) \(75 p=2600\)

Short answer

The inequality that can be used to find the minimum number of pianos to tune in order to meet business expenses is \(75p \geq 2600\). Therefore, the correct answer is (B).

Step-by-step solution

Understand the problem

Each piano tuning earns $75 and the monthly expenses are $2600. The goal is to find the minimum number of pianos to tune (p) in order to meet or exceed these expenses. This demonstrates a direct relation between the number of pianos tuned and the total monthly earnings. Looking at the inequalities given, option (C) can be disregarded as it shows the earnings decreasing with tuning more pianos, which contradicts the problem statement. Option (D) only provides the scenario where the income exactly equals the expenses. This leaves us with options (A) and (B).

Choose the correct inequality

When deciphering the inequalities, 'at least' infers 'greater than or equal to', representing that the earned amount should be more than or equal to the expenses. Hence, the correct inequality should be $75p \geq 2600$, which corresponds to option (B)

Key concepts

Algebraic Inequalities

Understanding algebraic inequalities is essential for comprehending many real-life scenarios where we deal with ranges instead of specific values. Inequalities are statements that outline the relative size or order of two values. They come in different forms: '<' means less than, '>' means greater than, '\(\leq\)' means less than or equal to, and '\(\geq\)' means greater than or equal to. In the case of the piano tuner exercise, we need an inequality that ensures the tuner's earnings are not less than the expenses.

The algebraic expression '\(75p\)' represents the total earnings from tuning '\(p\)' pianos. The inequality '\(75p \geq 2600\)' signifies that the earnings from tuning pianos must be greater than or equal to the monthly expenses of $2600 to maintain the business. This inequality correctly represents the situation and the tuner's requirement to at least break even on his expenses.

• Algebraic inequalities allow for a range of possible answers, not just one specific solution.
• They represent real-world situations where outcomes can vary within a certain limit.
• Inequalities can be simple, like in the piano tuner example, or they can be complex with variables on both sides.

Solving Inequalities

Solving inequalities involves finding all possible values of the variable that make the inequality true. Much like solving equations, one starts by isolating the variable. However, unlike equations, flipping the inequality sign is necessary when you multiply or divide by a negative number. Let's expand on the piano tuner example where the inequality '\(75p \geq 2600\)' needs to be solved for '\(p\)'.

Step-by-Step Solutions

To find the minimum number of pianos '\(p\)' necessary to meet expenses, we divide both sides of the inequality by 75:
\[\frac{75p}{75} \geq \frac{2600}{75}\]
\[p \geq \frac{2600}{75}\]
\[p \geq 34.67\]
Since you cannot tune a fraction of a piano, the minimum whole number of pianos the tuner must service to at least meet expenses is 35. Therefore, our solution for '\(p\)' is all integers greater than or equal to 35.

• Remember to perform the same operation on both sides of the inequality.
• Be cautious with negative numbers; these will invert the inequality sign.
• Pay attention to rounding: in the context of the problem, you might need to round up to the nearest whole number for a practical solution.

Linear Equations

Linear equations are foundational in algebra and represent a straight line when graphed on a coordinate plane. They take the standard form '\(Ax + By = C\)', where '\(A\)', '\(B\)', and '\(C\)' are constants, and '\(x\)' and '\(y\)' are variables. In the context of our piano tuner, a linear equation such as '\(75p=2600\)' (Option D from the multiple-choice question) would indicate that there is only one specific number of pianos that would result in the tuner's income exactly matching the expenses, without considering the possibility of earning more or servicing more pianos.

However, the real-world scenario implies the need for flexibility, hence why an inequality is more appropriate than a strict linear equation in this case. A linear equation provides one exact solution, but an inequality indicates a range of satisfactory answers. Understanding the differences between the two concepts is crucial for applying them correctly in various situations.

• Linear equations have a specific solution, whereas inequalities represent a set of possible solutions.
• The graph of a linear equation is a line, while the graph of an inequality shows a region in the coordinate plane.
• Real-world problems may require linear equations for precise answers or inequalities for ranges of answers.