Problem 14 - Prealgebra

Question

Write an inequality for each sentence. The race time of 86 minutes was greater than the winner's time.

Step-by-Step Solution

Answer

The inequality is \( 86 > t \).

Step 1: Understand the Variables

Identify the variables involved in the problem. Let the winner's time be represented by the variable \( t \), measured in minutes.

Step 2: Analyze the Inequality Condition

The sentence states that the race time of 86 minutes was greater than the winner's time. This translates to the inequality condition: 86 minutes > winner's time.

Step 3: Write the Inequality

Translate the inequality condition into a mathematical inequality. Using the variable for the winner's time, \( t \), the inequality becomes: \[ 86 > t \] .

Key Concepts

Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators that represent a particular value or set of values. They are a crucial part of understanding mathematical concepts, as they allow you to convey relationships and operations concisely. Consider the expression \[ 3 + 5 \]which simply means the addition of numbers 3 and 5. Now, in more complex mathematical expressions, you might see variables, which symbolize unknown values or quantities. For example, \[ x - 4 \]indicates that 4 is subtracted from some unknown value represented by \( x \). Expressions can be simple or complex, depending on the operations and components involved. A good grasp of how to read and construct them is essential for succeeding in mathematics.

Variables in Mathematics

Variables play a pivotal role in mathematics, especially when dealing with equations and inequalities. Variables are symbols used to represent unknown or changeable values. Commonly, letters like \( x \), \( y \), or \( t \) stand in place of numbers in mathematical problems. Imagine you're working with the sentence: "The race time of 86 minutes was greater than the winner's time." Here, the winner's time, an unknown value, can be aptly represented by the variable \( t \). This use of variables makes it simpler to deal with mathematical problems, as they provide a way to denote quantities that can vary, allowing for generalization and abstraction in problem-solving. Understanding how to use and manipulate variables is core to grasping deeper mathematical concepts.

Inequality Symbols

Inequality symbols are used to compare two values or expressions. They tell us whether the values are equal, or if one is larger or smaller than the other. The main inequality symbols include:

• \( > \) - Greater than
• \( < \) - Less than
• \( \geq \) - Greater than or equal to
• \( \leq \) - Less than or equal to

In the given exercise, we use the greater than symbol \( > \) because we know that the race time (86 minutes) is greater than the winner's time. Using inequality symbols helps you translate word problems into mathematical problems, paving the way for systematic problem-solving.

Problem Solving in Mathematics

Mathematical problem solving involves several steps to transform words or numerical information into solvable mathematical expressions or equations. The process generally includes:

• Understanding the problem by identifying known and unknown quantities.
• Representing these quantities with variables if necessary.
• Translating words into mathematical expressions or equations.
• Solving these equations to find the values of variables.
• Interpreting the results back in the context of the original problem.

For example, in our exercise, you start by understanding the context: "The race time of 86 minutes was greater than the winner's time." You identify that the winner's time is unknown and represent it as \( t \). Next, translate this situation into an inequality \( 86 > t \). Finally, this inequality tells you that the winner's time is any value less than 86 minutes. This structured approach is crucial when tasked with solving any mathematical problem.