One-Step Inequalities - Pre Algebra

1. A swimming pool is being drained at a constant rate. The amount of water remaining after $t$ hours is given by $500 - 25t$ gallons. For safety reasons, the pool must maintain more than $150$ gallons at all times. What is the maximum number of complete hours the pool can be drained?
   Answer: 13 hours We need $500 - 25t > 150$. Subtracting 500: $-25t > -350$. Dividing by $-25$ and flipping the inequality: $t < 14$. Since we want complete hours and $t$ must be less than 14, the maximum is 13 hours. Choice B would result in exactly 150 gallons (not more than). Choices C and D violate the constraint entirely.
2. A temperature monitoring system records that the laboratory temperature must stay below $-5°C$ to preserve certain samples. If the current temperature is $3x + 7$ degrees Celsius, which inequality represents the constraint on $x$?
   Answer: $x < -4$ The temperature $3x + 7$ must be less than $-5$, so we have $3x + 7 < -5$. Subtracting 7 from both sides gives $3x < -12$. Dividing by 3 gives $x < -4$. Choice B reverses the inequality sign incorrectly. Choice C results from adding 7 instead of subtracting. Choice D combines both errors.
3. A delivery truck can carry a maximum load of $2400$ pounds. It already has packages weighing $1650$ pounds loaded. If each additional package weighs $25$ pounds, what is the solution to the inequality representing how many more packages $p$ can be loaded?
   Answer: $p \leq 30$ and the truck can carry at most 30 more packages The inequality is $1650 + 25p \leq 2400$. Subtracting 1650: $25p \leq 750$. Dividing by 25: $p ≤ 30$. This means at most 30 packages. Choice A incorrectly states 'exactly' instead of 'at most'. Choice C uses strict inequality when equality is allowed. Choice D reverses the inequality direction completely.
4. A cell phone plan charges $35 per month plus $0.15 for each text message sent. If Jamie's monthly bill cannot exceed $50, what is the maximum number of text messages she can send?
   Answer: 100 text messages When you encounter a word problem involving a fixed cost plus a variable cost with a spending limit, you're looking at an inequality problem. You need to set up an equation that represents the total cost and find the maximum value that keeps you within the budget. Jamie's total monthly cost equals the fixed monthly charge plus the cost per text message times the number of messages: $35 + 0.15x \leq 50$, where $x$ is the number of text messages. To solve this inequality, subtract 35 from both sides: $0.15x \leq 15$. Then divide both sides by 0.15: $x \leq 100$. This means Jamie can send at most 100 text messages while staying within her $50 budget. Let's verify: $35 + 0.15(100) = 35 + 15 = 50$, which exactly meets her limit. Choice A (99 text messages) is incorrect because it's less than the maximum possible. While 99 messages would cost $35 + 0.15(99) = 49.85$, Jamie could actually afford one more message. Choice B (101 text messages) exceeds the budget: $35 + 0.15(101) = 50.15$, which is 15 cents over her limit. Choice D (333 text messages) represents a common error where students might divide 50 by 0.15, forgetting to subtract the fixed monthly charge first. Remember: when solving real-world inequality problems, always check whether your answer makes practical sense by substituting back into the original constraint. The maximum value should satisfy the inequality without exceeding the limit.
5. Maria has $45 to spend on art supplies. After buying a sketchbook for $12, she wants to buy markers that cost $5.50 each. What is the maximum number of markers she can buy?
   Answer: 6 markers After buying the sketchbook, Maria has $45 - 12 = 33$ dollars left. If she buys $m$ markers, she needs $5.50m \leq 33$. Dividing by 5.50 gives $m \leq 6$. Since $m$ must be a whole number, the maximum is 6 markers. Choice A ignores that she can afford exactly 6. Choice C would cost $7 × 5.50 = 38.50 > 33$. Choice D would cost $8 × 5.50 = 44 > 33$.
6. Which of the following represents the solution set for $\frac{x}{-3} > 4$ when graphed on a number line?
   Answer: Open circle at $-12$, shaded to the left toward negative numbers When solving inequalities involving fractions with negative denominators, you need to carefully track how the inequality sign changes during your algebraic steps. To solve $\frac{x}{-3} > 4$, multiply both sides by $-3$. Since you're multiplying by a negative number, you must flip the inequality sign. This gives you: $x < 4 \times (-3)$, which simplifies to $x < -12$. The solution $x < -12$ means all numbers less than $-12$. On a number line, this requires an open circle at $-12$ (since $x$ cannot equal $-12$) with shading extending left toward more negative numbers. This matches choice C. Let's examine why the other options are incorrect: Choice A places the circle correctly at $-12$ but shades right toward positive numbers, which would represent $x > -12$ instead of $x < -12$. Choice B uses a closed circle, which would only be correct if the original inequality included "or equal to" (≥ or ≤). Since we have a strict inequality (>), the circle must be open. Choice D places the circle at positive $12$ instead of negative $12$, likely resulting from forgetting to multiply $4$ by the negative sign. Remember this key rule: whenever you multiply or divide both sides of an inequality by a negative number, always flip the inequality sign. This is the most common mistake students make with inequality problems, so double-check your work by substituting a test value from your solution region back into the original inequality.
7. A bakery needs to package at least $180$ cupcakes for a large order. They have boxes that hold $15$ cupcakes each and have already filled $8$ boxes. The inequality $15(8 + b) \geq 180$ represents this situation, where $b$ is additional boxes needed. What is the minimum value of $b$?
   Answer: $b \geq 4$, so the minimum is 4 boxes When you encounter inequality word problems, start by understanding what the inequality represents and then solve it step by step to find what the variable must satisfy. Let's solve the inequality $15(8 + b) \geq 180$. First, divide both sides by 15: $8 + b \geq 12$. Then subtract 8 from both sides: $b \geq 4$. This means the bakery needs at least 4 additional boxes beyond the 8 they already filled. Let's verify: With 4 additional boxes, they'd have $8 + 4 = 12$ total boxes, containing $15 \times 12 = 180$ cupcakes, which exactly meets their requirement. Since they need "at least" 180 cupcakes, 4 additional boxes is the minimum. Choice A incorrectly states $b \geq 8$, which would mean they need 8 additional boxes on top of the 8 already filled—that's 16 total boxes for 240 cupcakes, way more than necessary. Choice B claims $b \geq 12$, suggesting 12 additional boxes for 20 total boxes and 300 cupcakes—again, excessive. Choice C states $b \geq 0$, which would mean no additional boxes are needed, but 8 boxes only hold 120 cupcakes, falling short of the 180 requirement. When solving inequality word problems, always substitute your answer back into the original context to verify it makes sense. Here, 4 additional boxes gives exactly 180 cupcakes, confirming this is indeed the minimum needed to satisfy "at least 180."

Start this quiz