A Limited Resource?...
A charity has a limited budget to provide food and shelter to homeless people. If x is the number of people to be helped and y is the number of days they can be supported, the budget constraints can be modeled by the linear inequality 3x + 2y ≤ 900. How will the inequality change if the budget is increased to 1200? Graphically represent the change and provide a justification for the same.
1 Answer
📌 CONCEPT: A linear inequality is a mathematical expression that compares two expressions using a less than or equal to (≤) or greater than or equal to (≥) symbol, indicating a constraint or a limit. In this case, the charity's budget constraint is modeled by the inequality 3x + 2y ≤ 900.
📐 RULE / FORMULA: To change the inequality when the budget is increased, we need to replace the original budget limit with the new budget limit, which is 1200. Therefore, the new inequality will be 3x + 2y ≤ 1200.
💡 WORKED EXAMPLE: Let's say the charity initially had a budget of 900 and they decided to increase it to 1200. To find the new inequality, we simply replace 900 with 1200 in the original inequality: 3x + 2y ≤ 1200. Graphically, this means the new line will be parallel to the original line but shifted 300 units upwards, as 1200 - 900 = 300.
⚠️ COMMON MISTAKE: Students may incorrectly assume that the slope of the line representing the inequality will change when the budget is increased, but in reality, the slope remains the same and only the y-intercept changes.
18 Aug 26
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