Problem-Solving Perseverance - Pre Algebra

1. Marcus is planning a school fundraiser selling two types of items: keychains for $3 each and bookmarks for $2 each. He needs to raise at least $240 and wants to sell exactly 100 items total. If he sells $k$ keychains, which inequality represents the constraint for reaching his fundraising goal, and what is the minimum number of keychains he must sell?
   Answer: $3k + 2(100-k) \geq 240$; he must sell at least 40 keychains The inequality $3k + 2(100-k) \geq 240$ correctly represents the constraint where Marcus sells k keychains at $3 each and (100-k) bookmarks at $2 each. Solving: $3k + 200 - 2k \geq 240$, so $k \geq 40$. Choice B uses the correct inequality but miscalculates the solution. Choice C switches the prices. Choice D incorrectly assumes all non-keychains cost $0.
2. A rectangular garden has length $(2x + 5)$ feet and width $(x - 3)$ feet. The gardener wants the area to be exactly 84 square feet. After setting up the equation $(2x + 5)(x - 3) = 84$ and expanding to get $2x^2 - x - 15 = 84$, what should be the next strategic step to solve this problem efficiently?
   Answer: Move 84 to the left side to get $2x^2 - x - 99 = 0$ and look for rational roots using the rational root theorem After getting $2x^2 - x - 99 = 0$, the rational root theorem is most efficient since potential rational roots are limited to factors of 99 divided by factors of 2. Testing shows $x = 9$ works. Choice A uses the quadratic formula unnecessarily when simpler methods exist. Choice B incorrectly factors (the correct factorization of $2x^2 - x - 15$ is $(2x - 5)(x + 3)$, but this doesn't help with the +84). Choice D is inefficient trial-and-error without systematic approach.
3. Tom is working on this problem: "The sum of three consecutive integers is 48. Find the integers." He sets up the equation $x + (x+1) + (x+2) = 48$ and solves to get $x = 15$. When he checks his work by substituting back, he gets $15 + 16 + 17 = 48$, which is correct. However, he notices that if the problem had asked for consecutive even integers instead, his method would need adjustment. What should his equation be for consecutive even integers with the same sum?
   Answer: $x + (x+2) + (x+4) = 48$ because consecutive even integers differ by 2 Consecutive even integers differ by 2, so if the first is x, the next two are (x+2) and (x+4). Choice A incorrectly suggests even integers differ by 1. Choice C uses an unnecessarily complicated representation - while 2x, (2x+2), (2x+4) are consecutive even integers, the simpler x, (x+2), (x+4) form works better where x represents the first even integer directly. Choice D incorrectly assumes the same equation works for different types of consecutive integers.
4. Sarah is solving a multi-step equation and gets stuck. She writes: "$2(x+3) - 5 = 3x - 7$. I distributed and got $2x + 6 - 5 = 3x - 7$, then $2x + 1 = 3x - 7$. Now I have $2x - 3x = -7 - 1$, so $-x = -8$, which means $x = 8$." When she checks her answer, it doesn't work. What should she do next to fix her error?
   Answer: Re-examine her sign handling because when moving terms she should have $2x - 3x = -7 + 1$ Sarah's error occurs when collecting terms. From $2x + 1 = 3x - 7$, subtracting $2x$ and adding $7$ to both sides gives $1 + 7 = 3x - 2x$, so $8 = x$. Her mistake was writing $2x - 3x = -7 - 1$ when it should be $2x - 3x = -7 + 1$ (since she's moving +1 to the right side). Choice A is wrong because her distribution is correct. Choice B incorrectly identifies the arithmetic error. Choice C identifies a consequence but not the root cause.
5. Carmen is solving the system: $\begin{cases} 2x + 3y = 12 \\ 4x + 6y = 20 \end{cases}$. She multiplies the first equation by -2 to get $-4x - 6y = -24$, then adds it to the second equation and gets $0 = -4$. She concludes the system has no solution. Her friend David says she made an error because the second equation should be $4x + 6y = 24$ to be consistent. Who is correct and why?
   Answer: Carmen is correct; the system as written is inconsistent and has no solution regardless of David's suggested change Carmen's analysis is mathematically sound. The system as given is inconsistent because the left sides of the equations are proportional (the second equation's left side is exactly twice the first), but the right sides are not proportional (24 ≠ 2×12 = 20). This creates a contradiction, so there is no solution. David's suggestion would create a system with infinitely many solutions, but that doesn't mean the original system was wrong - it could intentionally be inconsistent. Choice B misunderstands the mathematics. Choice C assumes there's a typo without justification. Choice D is mathematically impossible given the structure.
6. Jake is solving the inequality $-3(x - 4) < 2x + 8$. He distributes to get $-3x + 12 < 2x + 8$, then subtracts $2x$ from both sides: $-5x + 12 < 8$. Next, he subtracts 12 from both sides: $-5x < -4$. At this point, he needs to divide by $-5$. What must he remember to do, and what will his final answer be?
   Answer: Flip the inequality sign when dividing by a negative; final answer: $x > \frac{4}{5}$ When dividing both sides of an inequality by a negative number, the inequality sign must be flipped. From $-5x < -4$, dividing by $-5$ gives $x > \frac{4}{5}$. Choice B forgets to flip the sign. Choice C flips the sign correctly but gets the wrong final inequality direction. Choice D is completely incorrect - division by negative numbers is perfectly valid in inequalities, just requires flipping the sign.
7. Maria is planning a rectangular vegetable garden where the length is 4 feet more than twice the width. She has 60 feet of fencing to enclose the garden. After setting up her equation $2w + 2(2w + 4) = 60$ where $w$ is the width, she solves and gets $w = 8.67$ feet. However, she realizes this creates practical problems for her garden layout. What should she consider doing next to make this more practical?
   Answer: Round to $w = 8$ feet and use the leftover fencing for a separate small garden bed With $w = 8$ feet, the length becomes $2(8) + 4 = 20$ feet, and the perimeter is $2(8) + 2(20) = 56$ feet, leaving 4 feet of fencing unused - a practical solution. Choice A would require 62 feet of fencing (more than available). Choice C ignores the practical constraint that fractional feet are difficult to measure and work with in construction. Choice D unnecessarily complicates the problem by changing the constraint rather than working within it.

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