Creating Inequalities from Word Problems: A Practical Guide
When working through word problems, one of the key challenges is translating the language and conditions into mathematical expressions, particularly inequalities. This guide will walk you through the process of creating an inequality for a specific word problem, emphasizing the importance of accurately representing the conditions given.
Understanding the Problem
The given problem states that if Tom spends exactly $19.75, he will earn his free meal. If he spends less than $19.75, he will not get his free meal, and if he spends more than $19.75, he will earn his free meal. This problem revolves around the concept of inequalities and ensuring that the mathematical representation fully captures the conditions provided.
Initial Representation: $50 - 30.25 19.75
The equation provides the specific amount Tom needs to spend to earn his free meal. However, it does not directly form an inequality, and the key challenge is ensuring the representation includes all the conditions accurately.
Representing Tom’s Spending with Inequalities
The problem requires a representation that accounts for spending less than $19.75, spending exactly $19.75, and spending more than $19.75. Let's consider the different scenarios:
Spending Exactly $19.75
The inequality ( x 19.75 ) would represent the condition where Tom spends exactly $19.75. However, this is not an appropriate representation because it excludes the other conditions (spending less than $19.75 and spending more than $19.75).
Ensuring All Conditions Are Included
To accurately represent the conditions, we need to consider spending less than or more than $19.75. The optimal approach involves removing the exact value from consideration and creating an inequality that includes the permissible ranges.
Solution: ( x geq 19.74 )
To fully account for all conditions, the inequality ( x geq 19.74 ) is used. This inequality includes the exact amount of $19.75 and all amounts that are greater than $19.75. The reasoning behind this is straightforward:
If ( x 19.75 ), the condition is satisfied. If ( x > 19.75 ), the condition is also satisfied. If ( xConclusion
Creating inequalities from word problems requires a clear understanding of the conditions provided. In the case of the Tom's free meal problem, the key is to ensure that the inequality fully captures the requirements by including the necessary ranges. The inequality ( x geq 19.74 ) accurately represents the conditions and ensures that all possible valid spending amounts are included.