When approaching algebra, the ability to translate a real-world scenario into a mathematical model is the most critical skill. A word problem using a one step linear inequality focuses on defining a boundary condition rather than finding a single solution. This process involves identifying a variable, understanding the relationship described in the text, and converting that relationship into a mathematical statement.
Breaking Down the Language of Inequality
The first hurdle in solving a word problem is linguistic translation. You must look for specific keywords that dictate the direction of the mathematical symbol. Phrases like "at least," "no less than," and "minimum" indicate a greater than or equal to inequality ($\ge$). Conversely, terms such as "at most," "no more than," "below," and "under" signal a less than or equal to inequality ($\le$). Recognizing these cues immediately tells you whether the solution set lies to the right or left on a number line.
Identifying the Variable and Constant
Before writing the inequality, isolate the unknown quantity. Assign a variable, typically $x$, to represent the changing value in the scenario. The constant is the fixed number mentioned in the context of the problem. For example, if a problem states you need "at least 5 apples," the total number of apples ($x$) must be greater than or equal to 5. This translates to the inequality $x \ge 5$. Defining these two elements correctly ensures the mathematical statement accurately reflects the verbal description.
Imagine a scenario where a rental car company charges a flat fee of $50 plus $0.20 per mile. If you have a budget of $100, the problem requires you to determine how many miles you can drive. Let $m$ represent the miles driven. The total cost is represented by $0.20m + 50$. Since your budget is $100, the cost must be less than or equal to this amount. The one step linear inequality representing this situation is $0.20m + 50 \le 100$.
The Single Step Manipulation
Once the inequality is constructed, the solving phase mirrors solving an equation with one crucial difference regarding the inequality sign. The core objective is to isolate the variable on one side of the inequality. This is achieved by performing inverse operations. You may add, subtract, multiply, or divide to move the constant terms. However, if you multiply or divide by a negative number during any step, you must reverse the direction of the inequality symbol.
Returning to the rental car example, the first step subtracts 50 from both sides of the inequality. This simplifies the expression to $0.20m \le 50$. The next step involves dividing by the coefficient 0.20 to isolate $m$. Dividing by a positive number keeps the inequality direction the same, resulting in $m \le 250$. This tells you that the maximum number of miles you can drive without exceeding the budget is 250.
Interpreting the Solution Set
The final step in solving a word problem involving a one step linear inequality is interpreting the result in the context of the original problem. The answer $m \le 250$ means that any value less than or equal to 250 is valid. In practical terms, you cannot drive a negative number of miles, so the implicit domain is $0 \le m \le 250$. Understanding this range ensures the solution is realistic and applicable to the scenario, rounding out the problem-solving process.
