Question

In 1997 a software executive bid \(\$ 19,550\) for Superman's cape from the 1978 movie Superman. This was the highest bid. Let \(b\) represent the amount of a bid for the cape. Write an inequality for \(b\)

Short answer

The bid for the cape \(b\) would be represented by the inequality \(b \leq 19550\).

Step-by-step solution

Defining the terms

The task prompts to work with the variable \(b\) representing the amount of a bid for the cape. A specific reference is given stating that the highest bid was of $19,550 in 1997. This value needs to be defined in terms of \(b\).

Writing the Inequality

Since \(b\) represents a bid, any bid that is made should be less than or equal to the highest bid of $19,550. So, any bid would satisfy the condition that the bid \(b\) is less than or equal to $19,550. This can be represented as the inequality \(b \leq 19550\).

Interpretation of the inequality

The inequality \(b \leq 19550\) indicates that for any value of \(b\) to be possible, it shouldn’t be more than $19,550. In simpler terms, any bid for the cape would be either the highest bid, i.e., $19,550, or any value less than it.

Key concepts

Writing Inequalities

The art of writing inequalities is essential for translating real-world scenarios into mathematical expressions. An inequality, unlike an equation, shows a relationship where two values are not necessarily equal but rather one is either greater than, less than, equal to, or a combination of these with respect to the other.

In our Superman cape bidding example, the variable 'b' is used to express potential bids. Understanding that a bid can take any value up to the highest bid allows us to create the inequality that embodies this situation. We denote this by writing the inequality as \begin{align*}b &\leq 19550 \end{align*}, where the symbol \(\leq\) effectively captures the 'less than or equal to' relationship. This expression is concise yet rich with meaning, indicating the highest bid allowed and also implying all possible bids below that threshold.

When tasked to write inequalities, it's essential to identify the critical values or thresholds in the context provided and choose the appropriate inequality symbol to represent the real-world constraint mathematically.

Representing Variables

In algebra, representing variables is a fundamental concept. Variables act as placeholders for unknown or varying values and are often represented by letters of the alphabet. In the context of our example, the letter 'b' was chosen to signify the bid amount.

Choosing an appropriate variable is not arbitrary; it should make sense within the context. For instance, 'b' for 'bid' creates an intuitive connection, simplifying the process of understanding and remembering what the variable stands for. The choice of a suitable variable is a bridge between the language of mathematics and the real-world scenario it represents.

Once defined, the variable can be manipulated according to algebraic principles to solve problems or model situations. By giving a symbol to a certain quantity, like a bid, we can perform operations as if we were working with actual numbers, yet maintain a level of abstraction that applies to many potential values.

Interpreting Inequalities

Interpreting inequalities goes beyond just reading them. It involves a deeper understanding of the concepts and constraints they represent. Referring back to our Superman cape problem, the inequality \(b \leq 19550\) provides a boundary condition for what constitutes an acceptable bid. A bid can be anywhere from \(0 up to, and including, \)19,550.

In practical terms, interpretation means someone can bid \(5000, \)15000, or even \(19550, but not a penny more. It encapsulates the idea that bids higher than \)19,550 would be outside the acceptable range and thus, not a feasible outcome. The inequality tells a story: it sets a stage for all the possible scenarios within a given context and acts as a guidepost for decision-making.

As students interpret inequalities, they must learn to visualize the range of values an inequality covers—a skill that forms the bedrock of mathematical reasoning and problem-solving in various real-life situations beyond the classroom.