Problem 28 - Algebra 2 and Trigonometry

Question

A theater owner knows that to make a profit as well as to comply with fire regulations, the number of tickets that he sells can differ from 225 by no more than \(75 .\) How many tickets can the theater owner sell in order to make a profit and comply with fire regulations?

Step-by-Step Solution

Answer

The theater owner can sell between 150 and 300 tickets.

Step 1: Understanding the Problem

The problem states that the theater owner can sell a number of tickets that differ from 225 by no more than 75. This means the number of tickets sold must be within a range of 75 tickets either less than or more than 225.

Step 2: Translating into a Mathematical Inequality

We need to express the condition mathematically. If we let the number of tickets sold be represented by the variable \( x \), then the condition can be expressed as: \(|x - 225| \leq 75\).

Step 3: Solving the Absolute Value Inequality

The inequality \(|x - 225| \leq 75\) can be broken down into two separate inequalities: 1. \(x - 225 \leq 75\)2. \(x - 225 \geq -75\)

Step 4: Solving the First Inequality

For the inequality \(x - 225 \leq 75\), add 225 to both sides to isolate \(x\):\[x \leq 300\]

Step 5: Solving the Second Inequality

For the inequality \(x - 225 \geq -75\), add 225 to both sides to isolate \(x\):\[x \geq 150\]

Step 6: Determining the Range of Tickets

Combining the results from Steps 4 and 5, we find that the theater owner can sell between 150 and 300 tickets to make a profit and comply with regulations.

Key Concepts

Absolute Value

Understanding absolute value is crucial for solving many mathematical problems, especially those involving distance or deviation from a central point. In simple terms, the absolute value of a number is its distance from zero on a number line. The symbol used to denote absolute value is vertical bars, as in \(|x|\). For example, \(|5|\) is 5, and \[ |-5| \] is also 5 because both are 5 units away from zero. Absolute value helps in determining how much a value deviates from a specific number, disregarding direction. In other words, it's about magnitude without considering the sign.

• It's always non-negative.
• In equations, absolute values are used to define a range or tolerance level around a point.

In exercises where a limit of deviation or allowance is defined, absolute value is often used to effectively specify those limits.

Mathematical Inequality

Inequalities are similar to equations, but instead of expressing equal values, they define a relationship where one side is greater or less than the other. Inequalities can often be seen using symbols like \(>\), \(<\), \(\geq\) (greater than or equal), and \(\leq\) (less than or equal). These expressions are essential in describing constraints and limitations in various scenarios. In our exercise, the theater owner needs to abide by a limitation on the number of tickets. This is mathematically expressed as an inequality, specifically \(|x - 225| \leq 75\). Inequalities help to express that the number of tickets sold should always fall within a specific range, making sure the terms are kept as intended. By solving the inequality, you determine not just an isolated value but a range of acceptable values.

• They help in establishing boundaries.
• Provide a set of possible solutions rather than just one.

It's important to respect these conditions to ensure outcomes remain viable and regulations are met.

Range of Values

The range of values in a mathematical context indicates the span of values that fulfill a particular condition or set of conditions. In our theater ticket problem, determining this range is crucial for understanding the owner's boundaries within which he can operate. After solving the absolute value inequality \(|x - 225| \leq 75\), we find the range of values for \(x\) that satisfies both inequalities. These solutions, \(x \leq 300\) and \(x \geq 150\), offer the inclusive range of 150 to 300.

• It's the set of all possible values that a variable, such as the number of tickets in this case, can take.
• Ensures solutions remain practical within defined limits.

Understanding the range not only helps in planning and decision-making but also ensures all conditions are met effectively. It ensures precision in actions, like ticket sales, consistent with both profitability and regulatory compliance.