Question

a manager is given a budget of a maximum of $150 to purchase pens and pencils for the office. each pack of pens, x, contains 24 pens and costs $5.50. each pack of pencils, y, contains 40 pencils and costs $6.50. the manager wants to purchase at least 600 pens and pencils. which system of inequalities correctly models this scenario?

5.50x + 6.50y ≤ 15
24x + 40y ≥ 600

6.50x + 5.50y ≤ 15
24x + 40y ≥ 600

5.50x + 6.50y < 15
24x + 40y ≥ 600

5.50x + 6.50y ≤ 15
24x + 40y ≤ 600

Explanation:

Step1: Analyze the cost constraint

The budget is a maximum of $150. Each pack of pens (x) costs $5.50 and each pack of pencils (y) costs $6.50. So the total cost LXI0 must be less than or equal to 150? Wait, no, wait the first option has 15, maybe a typo? Wait, no, maybe the original problem had a typo, but looking at the options, the first inequality for cost: the total cost of pens and pencils should be ≤150? But the options have 15. Wait, maybe it's a typo and should be 150, but the first option's cost inequality is LXI1? No, that can't be. Wait, no, maybe the budget is $150, so the cost is \(5.50x + 6.50y \leq 150\), but the options have 15. Wait, maybe it's a mistake in the problem, but looking at the options, the first system: cost inequality \(5.50x + 6.50y \leq 15\) (maybe a typo, should be 150) and quantity inequality \(24x + 40y \geq 600\) (since each pen pack has 24 pens, x packs: 24x pens; each pencil pack has 40 pencils, y packs: 40y pencils. Total pens and pencils should be at least 600, so \(24x + 40y \geq 600\)). Now check the cost: pens cost $5.50 per pack (x), pencils $6.50 per pack (y). So total cost \(5.50x + 6.50y \leq 150\) (but options have 15, maybe a typo, but the first option's cost inequality is \(5.50x + 6.50y \leq 15\), but that's too low. Wait, maybe the budget is $15? No, the problem says $150. Wait, maybe the options have a typo, but among the options, the first system has cost inequality \(5.50x + 6.50y \leq 15\) (maybe a typo for 150) and quantity \(24x + 40y \geq 600\), which matches the quantity requirement (at least 600, so ≥600). The other options: second option has cost \(6.50x + 5.50y\) (swapped, which is wrong, since x is pens, cost 5.50, y is pencils, cost 6.50). Third option has \(<15\), which is strict, but budget is maximum, so ≤. Fourth option has quantity ≤600, which is wrong (needs at least 600, so ≥). So the first system: cost \(5.50x + 6.50y \leq 15\) (maybe typo, but among options, this is the only one with correct quantity inequality and cost inequality with correct coefficients (pens: 5.50x, pencils: 6.50y) and ≤ for cost, ≥ for quantity.

Step2: Confirm the quantity inequality

Each pack of pens (x) has 24 pens, so total pens: 24x. Each pack of pencils (y) has 40 pencils, so total pencils: 40y. Total pens and pencils: \(24x + 40y\). The manager wants at least 600, so \(24x + 40y \geq 600\), which matches the first system's second inequality. The cost: each pen pack is $5.50, so x packs cost LXI2; each pencil pack is $6.50, so y packs cost \(6.50y\). Total cost: \(5.50x + 6.50y\), which must be ≤150 (but option has 15, maybe typo). Among the options, the first system has the correct quantity inequality and correct cost coefficients (pens: 5.50x, pencils: 6.50y) with ≤ for cost. The other options: second has cost coefficients swapped, third has strict inequality, fourth has quantity ≤. So the first system is correct.

Answer:

The first option:
\(5.50x + 6.50y \leq 15\)
\(24x + 40y \geq 600\)