Creating/Solving One Variable Equations/Inequalities - Algebra 1

Movie tickets cost $12 each, and you have $85 to spend. You also want to buy one large popcorn for $7. What is the maximum number of tickets you can afford if you also buy the large popcorn?
Answer: Maximum $t$ is $6$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. For inequalities, words like 'at most,' 'maximum,' 'no more than' signal ≤ (less than or equal), while 'at least,' 'minimum,' 'no less than' signal ≥ (greater than or equal). 'More than' means > (strict), and 'less than' means <. These key phrases tell you which inequality symbol to use! The context 'Movie tickets cost $12 each, and you have $85 to spend. You also want to buy one large popcorn for $7' uses the phrase 'have $85 to spend,' which signals ≤. Setting up: 12t + 7 ≤ 85. Solving: subtract 7 → 12t ≤ 78, divide by 12 → t ≤ 6.5. This means maximum of 6 whole tickets, since you can't buy a fraction of a ticket. Choice A is correct because it properly sets up the inequality from context, solves correctly, and interprets appropriately, giving maximum t=6 whole tickets. Choice B has the inequality symbol backwards or ignores the popcorn: perhaps doing 12t ≤ 85 → t ≤ 7.08 (to 7), but 'have $85 to spend' including popcorn means ≤ after adding $7, not ≥ or excluding it. These phrases can be tricky—memorizing which direction they go really helps! For inequalities, make a quick reference card: 'at most/maximum/no more than' → ≤ (can equal or be less), 'at least/minimum/no less than' → ≥ (can equal or be more), 'more than/over' → > (strictly greater), 'less than/under' → < (strictly less). Having these memorized means you'll never use the wrong symbol!
A rectangular garden has a length that is 5 feet more than its width. The area is 84 square feet. Write and solve an equation to find the width of the garden. Let $w$ = the width (in feet).
Answer: $w = 7$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Different contexts lead to different equation types: constant rates give linear equations (like cost = rate × quantity + fee), area problems often give quadratics (like length × width = area), and growth over time gives exponentials (like population = initial × (growth rate)^time). The context clues tell you which form to use. This is an area problem, which means quadratic! Let w = width. The relationship 'length is 5 more than width, area 84' translates to w(w + 5) = 84, or w² + 5w - 84 = 0. Using quadratic formula, discriminant 25 + 336 = 361 = 19², so w = [-5 ± 19]/2, giving w = 7 or w = -12 (discard negative). So valid solution w = 7 with interpretation: the garden is 7 feet wide and 12 feet long. Choice A is correct because it properly sets up the quadratic from the area context, solves correctly, and interprets appropriately, giving w = 7 feet that makes sense. Choice C solves the equation correctly but doesn't check the context: w = 12 would imply length = 17, but that's swapping variables— the question asks for width, which is the smaller one. Always ask: does my answer make sense in the real world? This catches a lot of mistakes! The 'reality check' is your best friend: after solving, substitute your answer back into the original equation (math check), then ask 'does this make sense?' (reality check). Can dimensions be negative? Can you buy 7.3 shirts? Can there be -4 hours? The context tells you what's possible and what's not!
A ball is thrown upward from the ground. Its height (in feet) after $t$ seconds is given by $h=-16t^2+64t$ . Solve for the time(s) when the ball is on the ground again (when $h=0$ ). Let $t$ = time in seconds.
Answer: $t=0$ or $t=4$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Different contexts lead to different equation types: constant rates give linear equations (like cost = rate × quantity + fee), area problems often give quadratics (like length × width = area), and growth over time gives exponentials (like population = initial × (growth rate)^time). The context clues tell you which form to use. This is a motion problem, which means quadratic! The ball's height equation is h = -16t² + 64t. When the ball is on the ground, h = 0, so we need to solve -16t² + 64t = 0. Factoring: t(-16t + 64) = 0, which gives us t = 0 or -16t + 64 = 0. From the second equation: t = 4. Interpreting: the ball starts on the ground at t = 0 seconds and returns to the ground at t = 4 seconds. Choice B is correct because it finds both solutions to the quadratic equation and interprets them correctly: t = 0 (initial launch) and t = 4 (landing). Choice A gives only t = 2, which would be when the ball reaches maximum height (vertex of the parabola), not when it's on the ground. For quadratics in physics contexts, h = 0 often has two meaningful solutions! Different contexts = different equation types: constant rates and simple relationships → linear, area and projectile motion → quadratic, working together and rate problems → rational, growth/decay over time → exponential. Recognizing these patterns helps you set up the right type of equation immediately!
A store offers a membership where customers pay $30 annually and then receive a 15% discount on all purchases. Without membership, customers pay full price. For what annual spending amount $x$ (in dollars) would the membership cost exactly break even with non-membership shopping?
Answer: $30 + 0.85x = x$ When you encounter break-even problems, you need to set up an equation where the total costs of both options are equal. Here, you're comparing the total annual cost of membership shopping versus non-membership shopping. With membership, you pay $30$ upfront plus 85% of your purchases (since you get a 15% discount, you pay 100% - 15% = 85%). So the total cost is $30 + 0.85x$ . Without membership, you simply pay the full price $x$ for your purchases. At the break-even point, these costs are equal: $30 + 0.85x = x$ . This is answer choice D. Let's examine why the other options are incorrect: Choice A ( $0.85x = x - 30$ ) incorrectly suggests that your discounted purchases equal your full purchases minus $30$ . This doesn't account for the membership fee you actually paid. Choice B ( $30 + 0.15x = x$ ) mistakenly uses 0.15 as what you pay, but 0.15 represents your savings, not your payment. You pay 85% of the original price, not 15%. Choice C ( $30 = 0.15x$ ) only considers the membership fee equaling your savings, ignoring that you still have to pay for the discounted items. Study tip: In discount problems, always identify what percentage you actually pay (100% minus the discount percentage) rather than the discount itself. Set up break-even equations by making total costs equal, ensuring you include all fees and payments for each option.
A rental car company charges a flat fee of $25 plus $0.15 per mile driven. If Marcus wants to spend at most $85 on his rental, which inequality correctly represents the maximum number of miles $m$ he can drive?
Answer: $25 + 0.15m \leq 85$ The total cost is the flat fee ($25) plus the per-mile charge ($0.15m). Since Marcus wants to spend 'at most' $85, the total cost must be less than or equal to $85, giving us $25 + 0.15m ≤ 85$ . Choice B uses the wrong inequality direction (≥ instead of ≤). Choice C incorrectly subtracts the flat fee instead of adding it. Choice D incorrectly multiplies the flat fee by miles and treats the per-mile rate as a constant.
The height of a ball thrown upward can be modeled by $h(t) = -16t^2 + 48t + 6$ , where $h$ is the height in feet and $t$ is the time in seconds. Which equation would you solve to find when the ball reaches a height of 54 feet?
Answer: $-16t^2 + 48t + 6 = 54$ To find when the ball reaches a height of 54 feet, we set the height function equal to 54: $-16t^2 + 48t + 6 = 54$ . Choice B incorrectly moves 54 to the left side and 6 to the right side. Choice C omits the initial height term (+6). Choice D incorrectly changes the sign of the initial height term to negative.
A streaming service charges $9 per month plus a one-time sign-up fee of $15. If your total cost for the first month is $42, which equation can be used to find $m$ , the number of months you have paid for?
Answer: $9m+15=42$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Creating an equation from a word problem means identifying the unknown quantity (your variable m), finding what you know (the numbers), and writing an equation that captures the relationship the problem describes. For example, 'a plumber charges $50 plus $40 per hour for a $170 job' becomes 50 + 40h = 170, where h is hours worked. Let's break down 'streaming service charges': We need to find the number of months paid for, so let m = number of months. From the context, you pay $9 per month plus a one-time sign-up fee of $15, and the total for the first month is $42. This means 9m + 15 = 42. Note that if the total is for the first month only, then m = 1, but we're asked for the equation to find m. Choice A is correct because it properly sets up the equation from context: monthly charge times months plus one-time fee equals total cost, giving 9m + 15 = 42. Choice B has the coefficients reversed: it multiplies the one-time fee by m and adds the monthly charge once, which doesn't match the problem's structure. When the context says 'per month,' that's what gets multiplied by the number of months! Different contexts = different equation types: constant rates and simple relationships → linear, area and projectile motion → quadratic, working together and rate problems → rational, growth/decay over time → exponential. Recognizing these patterns helps you set up the right type of equation immediately!
A video game costs $60 before tax. A store coupon takes 25% off the original price. Write and solve an equation to find the sale price. Let $p$ = the sale price (in dollars).
Answer: $p = 45$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Creating an equation from a word problem means identifying the unknown quantity (your variable x), finding what you know (the numbers), and writing an equation that captures the relationship the problem describes. For example, 'a plumber charges $50 plus $40 per hour for a $170 job' becomes 50 + 40h = 170, where h is hours worked. Let's break down 'video game $60, 25% off, find sale price p': We need to find the sale price, so let p = sale price. From the context, sale price is original minus discount, so p = 60 - 0.25*60, or equivalently p = 60*0.75. Solving: 60*0.75 = 45 → p = 45. In context: the sale price is $45. Choice C is correct because it properly sets up the equation with the percentage discount and solves correctly, giving p=45 dollars. Choice D solves for the original but the question asks for sale price: after finding discount=15, subtracting from 60 gives 45, not stopping at 75% or something. Read the question carefully to see exactly what they want! For percent problems (discount, markup, tax), remember: discount/off means subtract (original - discount = sale price), markup/increase/tax means add (original + increase = new price). Common setup: 'After 20% off, price is $64' means 0.80 × original = 64 (because you pay 80% after 20% off). The multiplication happens before setting equal to the final price!
Two pumps fill a tank together in 4 hours. Pump A alone can fill the tank in 6 hours. Write and solve an equation to find how long Pump B would take to fill the tank alone. Let $x$ = the number of hours for Pump B alone.
Answer: $x = 12$ hours This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Different contexts lead to different equation types: constant rates give linear equations (like cost = rate × quantity + fee), area problems often give quadratics (like length × width = area), and growth over time gives exponentials (like population = initial × (growth rate)^time). The context clues tell you which form to use. For rate problems like this, we set up: rates add for working together, so (1/6) + (1/x) = 1/4. This gives us rational equation. Solving: 1/x = 1/4 - 1/6 = (3-2)/12 = 1/12, so x=12. Interpreting: Pump B takes 12 hours alone to fill the tank. Choice C is correct because it properly sets up the rational equation from the work rates context, solves correctly, and interprets appropriately, giving x=12 hours. Choice B makes an arithmetic error: perhaps subtracting rates incorrectly as 1/6 - 1/4 = -1/12, but it's together rate minus A gives B. With all the steps in solving word problems—setting up, solving, interpreting—it's easy for calculation errors to slip in. Double-checking arithmetic is always worth it! The foolproof word problem strategy: (1) Read carefully and identify what's unknown—that's your variable, (2) Find what you know—those are your numbers, (3) Look for relationships—how are quantities connected? This gives you the equation, (4) Solve the equation using appropriate methods, (5) Check: does your answer satisfy the equation AND make sense in context? Following these steps systematically prevents most mistakes!
Pump A can fill a pool in 12 hours. Pump B can fill the same pool in $x$ hours. Working together, they fill the pool in 8 hours. Set up and solve an equation to find how long Pump B takes to fill the pool alone. Let $x$ = Pump B's time (in hours).
Answer: $x = 24$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Different contexts lead to different equation types: constant rates give linear equations (like cost = rate × quantity + fee), area problems often give quadratics (like length × width = area), and growth over time gives exponentials (like population = initial × (growth rate)^time). The context clues tell you which form to use. For rate problems like this, we set up: rates add when working together, so Pump A rate 1/12 + Pump B rate 1/x = combined rate 1/8. This gives us 1/12 + 1/x = 1/8. Solving: subtract 1/12 → 1/x = 1/8 - 1/12 = (3-2)/24 = 1/24, so x=24. Interpreting: Pump B takes 24 hours alone. Choice B is correct because it properly sets up the rational equation from context, solves correctly, and interprets appropriately, giving x=24 hours. Choice A makes an arithmetic error: perhaps in subtracting fractions, doing 1/8 - 1/12 = (3-2)/24=1/24 correctly but then misinterpreting. With all the steps in solving word problems—setting up, solving, interpreting—it's easy for calculation errors to slip in. Double-checking arithmetic is always worth it! The foolproof word problem strategy: (1) Read carefully and identify what's unknown—that's your variable, (2) Find what you know—those are your numbers, (3) Look for relationships—how are quantities connected? This gives you the equation, (4) Solve the equation using appropriate methods, (5) Check: does your answer satisfy the equation AND make sense in context? Following these steps systematically prevents most mistakes!
The number of members in a chess club can be represented by $\frac{180}{n + 5}$ , where $n$ is the number of years since the club started. Which inequality represents when the club will have fewer than 12 members?
Answer: $\frac{180}{n + 5} < 12$ When you encounter a word problem involving inequalities, your first step is to translate the English phrase into mathematical language. The key phrase here is "fewer than 12 members." Since the number of members is represented by $\frac{180}{n + 5}$ , and we want this to be fewer than 12, we need the inequality $\frac{180}{n + 5} < 12$ . The phrase "fewer than" directly translates to the less-than symbol (<). Looking at the wrong answers: Choice A gives us $180 < 12(n + 5)$ , which comes from incorrectly cross-multiplying the correct inequality but then forgetting to include the fraction form. Choice B shows $\frac{180}{n + 5} > 12$ , which uses the wrong inequality symbol—this would represent when the club has more than 12 members, not fewer. Choice C presents $\frac{12}{n + 5} < 180$ , which flips the numerators incorrectly; this doesn't represent the given situation at all since 12 should not be in the numerator. The correct answer is D: $\frac{180}{n + 5} < 12$ . This directly states that the number of members (the entire expression $\frac{180}{n + 5}$ ) is less than 12. Study tip: When translating word problems to inequalities, write out the English phrase first, then substitute the mathematical expressions. "Fewer than," "less than," and "under" all translate to <, while "more than," "greater than," and "exceeds" translate to >.
A rectangular garden has a length that is 5 feet more than its width. The area of the garden is 84 square feet. Write and solve an equation to find the width. Let $w$ = the width in feet.
Answer: $w=7$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Different contexts lead to different equation types: constant rates give linear equations (like cost = rate × quantity + fee), area problems often give quadratics (like length × width = area), and growth over time gives exponentials (like population = initial × (growth rate)^time). The context clues tell you which form to use. This is an area problem, which means quadratic! Let w = width in feet. The relationship 'length is 5 feet more than width' translates to length = w + 5. Since area = length × width = 84, we get w(w + 5) = 84. Expanding: w² + 5w = 84, so w² + 5w - 84 = 0. Using factoring: (w + 12)(w - 7) = 0, we get w = -12 or w = 7. But wait—checking context: width can't be negative! So w = 7 feet with interpretation: the garden is 7 feet wide and 12 feet long. Choice A is correct because it properly sets up the quadratic equation from the area context, solves correctly, and eliminates the extraneous negative solution, giving w = 7 feet. Choice B would give a length of 14 feet and area of 9 × 14 = 126 square feet, not 84. When solving quadratics from real-world contexts, always check both solutions—one often doesn't make physical sense! The 'reality check' is your best friend: after solving, substitute your answer back into the original equation (math check), then ask 'does this make sense?' (reality check). Can dimensions be negative? Can you buy 7.3 shirts? Can there be -4 hours? The context tells you what's possible and what's not!
A ball is thrown upward from the ground. Its height (in feet) after $t$ seconds is given by $h = -16t^2 + 64t$ . Solve for the time(s) when the ball is on the ground. Let $t$ = time in seconds, and set $h = 0$ .
Answer: $t = 0$ and $t = 4$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Different contexts lead to different equation types: constant rates give linear equations (like cost = rate × quantity + fee), area problems often give quadratics (like length × width = area), and growth over time gives exponentials (like population = initial × (growth rate)^time). The context clues tell you which form to use. This is a motion problem, which means quadratic! Let t = time. The relationship 'height h = -16t² + 64t, find when on ground' translates to set h=0: -16t² + 64t = 0. Factoring: -16t(t - 4) = 0, so t=0 or t=4. Checking context: both make sense as start and landing times. So valid solutions t=0 and t=4 with interpretation: the ball is on the ground at launch and after 4 seconds. Choice B is correct because it properly sets up the quadratic from the context, solves correctly finding both roots, and interprets appropriately, giving the times when height is zero. Choice D finds one solution to the quadratic but misses the other (or keeps only one). When solving t² - 4t = 0, we get two values: 0 and 4. Checking context: both are valid for projectile motion! For quadratics, always check both solutions against the real-world situation! The 'reality check' is your best friend: after solving, substitute your answer back into the original equation (math check), then ask 'does this make sense?' (reality check). Can dimensions be negative? Can you buy 7.3 shirts? Can there be -4 hours? The context tells you what's possible and what's not!
A ball is thrown upward from the ground. Its height (in feet) after $t$ seconds is given by $h = -16t^2 + 64t$ . Solve for the time(s) when the ball is on the ground. Let $t$ = time in seconds.
Answer: $t = 0$ and $t = 4$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. Different contexts lead to different equation types: constant rates give linear equations (like cost = rate × quantity + fee), area problems often give quadratics (like length × width = area), and growth over time gives exponentials (like population = initial × (growth rate)^time). The context clues tell you which form to use. This is a motion problem, which means quadratic! Let t = time in seconds. The relationship 'height after t seconds is given by h = -16t² + 64t' and 'when the ball is on the ground' translates to set h=0: -16t² + 64t = 0. Using factoring, -16t(t - 4) = 0, so t=0 or t=4. Checking context: both make sense—at t=0 (start) and t=4 (lands). Interpreting: the ball is on the ground at 0 seconds and after 4 seconds. Choice B is correct because it properly sets up the quadratic equation from context, solves correctly, and interprets appropriately, giving t=0 and t=4 seconds. Choice D finds one solution to the quadratic but misses the other: solving gives two values, but perhaps they ignored t=0 since it's the start. For quadratics, always check both solutions against the real-world situation! The 'reality check' is your best friend: after solving, substitute your answer back into the original equation (math check), then ask 'does this make sense?' (reality check). Can dimensions be negative? Can you buy 7.3 shirts? Can there be -4 hours? The context tells you what's possible and what's not!
You have at most $50 to spend on notebooks that cost $4 each. Write an inequality representing this constraint and find the maximum number of notebooks you can buy. Let $n$ = number of notebooks.
Answer: Maximum $n=12$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. For inequalities, words like 'at most,' 'maximum,' 'no more than' signal ≤ (less than or equal), while 'at least,' 'minimum,' 'no less than' signal ≥ (greater than or equal). 'More than' means > (strict), and 'less than' means <. These key phrases tell you which inequality symbol to use! The context 'at most $50 to spend' uses the phrase 'at most,' which signals ≤. Setting up: cost of notebooks ≤ money available, so 4n ≤ 50. Solving: n ≤ 12.5. This means a maximum of 12 whole notebooks (can't buy half a notebook!). Choice B is correct because it properly sets up the inequality from context, solves correctly, and interprets appropriately in the real world, giving maximum n = 12 notebooks. Choice C would cost 4(13) = $52, which exceeds the $50 budget. When dealing with discrete items like notebooks, always round down to stay within the constraint! For inequalities, make a quick reference card: 'at most/maximum/no more than' → ≤ (can equal or be less), 'at least/minimum/no less than' → ≥ (can equal or be more), 'more than/over' → > (strictly greater), 'less than/under' → < (strictly less). Having these memorized means you'll never use the wrong symbol!
A jacket is on sale for 20% off. The sale price is $72. Write and solve an equation to find the original price of the jacket. Let $p$ = the original price (in dollars).
Answer: $p = 90$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. For percent problems (discount, markup, tax), remember: discount/off means subtract (original - discount = sale price), markup/increase/tax means add (original + increase = new price). Common setup: 'After 20% off, price is $72' means 0.80 × original = 72 (because you pay 80% after 20% off). The multiplication happens before setting equal to the final price! Let's break down 'A jacket is on sale for 20% off. The sale price is $72': We need to find original price, so let p = original price. From the context, sale price is 80% of original means 0.8p = 72. Solving: divide both sides by 0.8 → p = 72 / 0.8 = 90. In context: the original price was $90. Choice B is correct because it properly sets up the equation from context and solves correctly, giving p=90 dollars. Choice A makes an arithmetic error or reverses the percent: perhaps adding 20% to 72 (72 × 1.2 = 86.4), but to find original, you divide by 0.8, not multiply by 1.2. With all the steps in solving word problems—setting up, solving, interpreting—it's easy for calculation errors to slip in. Double-checking arithmetic is always worth it! After solving, always check two things: (1) Does your answer satisfy the equation when you substitute back? (2) Does it make sense in the real world? You can't have -3 people or 7.5 shirts if buying whole shirts! The context puts restrictions on valid solutions.
Tickets to a school play cost $9 each, and there is a one-time online fee of $4 per order. Jordan has at most $40 to spend. Write an inequality representing the number of tickets Jordan can buy. Let $t$ = the number of tickets.
Answer: $9t + 4 \le 40$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. For inequalities, words like 'at most,' 'maximum,' 'no more than' signal ≤ (less than or equal), while 'at least,' 'minimum,' 'no less than' signal ≥ (greater than or equal). 'More than' means > (strict), and 'less than' means <. These key phrases tell you which inequality symbol to use! The context 'tickets cost $9 each with a $4 fee, at most $40' uses the phrase 'at most,' which signals ≤. Setting up: cost per ticket times number plus fee ≤ total available, so 9t + 4 ≤ 40. Solving: subtract 4 → 9t ≤ 36, divide by 9 → t ≤ 4. This means Jordan can buy a maximum of 4 whole tickets. Choice A is correct because it properly sets up the inequality from the context with the fee added and uses ≤ for 'at most,' giving t ≤ 4 tickets within budget. Choice D sets up the inequality incorrectly: it switches the variables to 4t + 9 ≤ 40, which would be like $4 per ticket and $9 fee, but the problem says $9 tickets and $4 fee—reading carefully for relationships is key! For inequalities, make a quick reference card: 'at most/maximum/no more than' → ≤ (can equal or be less), 'at least/minimum/no less than' → ≥ (can equal or be more), 'more than/over' → > (strictly greater), 'less than/under' → < (strictly less). Having these memorized means you'll never use the wrong symbol!
A phone plan costs $25 per month plus $0.10 per text message. Your monthly budget for the plan is at most $40. What is the maximum number of text messages you can send? Let $t$ = number of text messages.
Answer: $t\le 150$ This question tests your ability to translate a real-world situation into a mathematical equation or inequality, solve it, and interpret the result in the original context. For inequalities, words like 'at most,' 'maximum,' 'no more than' signal ≤ (less than or equal), while 'at least,' 'minimum,' 'no less than' signal ≥ (greater than or equal). 'More than' means > (strict), and 'less than' means <. These key phrases tell you which inequality symbol to use! The context 'budget is at most $40' uses the phrase 'at most,' which signals ≤. Setting up: monthly cost + text cost ≤ budget, so $25 + 0.10t \leq 40$ . Solving: $0.10t \leq 15$ , so $t \leq 150$ . This means a maximum of 150 text messages. Choice B is correct because it properly sets up the inequality from context and solves correctly, giving $t \leq 150$ text messages maximum. Choice A incorrectly solves the inequality: after subtracting 25 from both sides to get $0.10t \leq 15$ , dividing by 0.10 gives $t \leq 150$ , not $t \leq 250$ . This appears to be an arithmetic error in the division step. The foolproof word problem strategy: (1) Read carefully and identify what's unknown—that's your variable, (2) Find what you know—those are your numbers, (3) Look for relationships—how are quantities connected? This gives you the equation, (4) Solve the equation using appropriate methods, (5) Check: does your answer satisfy the equation AND make sense in context? Following these steps systematically prevents most mistakes!
A bacteria population doubles every 3 hours. If the initial population is 500 bacteria, which equation can be used to find the time $t$ (in hours) when the population first exceeds 10,000?
Answer: $500 \cdot 2^{t/3} > 10{,}000$ Since the population doubles every 3 hours, after $t$ hours there have been $t/3$ doubling periods. The population is $500 \cdot 2^{t/3}$ , and we need this to exceed 10,000. Choice B uses $2^{3t}$ which would mean the population doubles 3 times per hour instead of once every 3 hours. Choice C uses linear growth instead of exponential growth. Choice D incorrectly uses base 3 and reverses the fraction in the exponent.
The temperature $T$ (in degrees Celsius) of a cooling object after $t$ minutes follows the equation $T = 80 \cdot (0.95)^t + 20$ . Which equation would you solve to find when the temperature drops to exactly 35°C?
Answer: $80 \cdot (0.95)^t + 20 = 35$ When you encounter exponential equations in real-world contexts, you need to set up an equation where the given formula equals the target value you're looking for. Here, you have the temperature formula $T = 80 \cdot (0.95)^t + 20$ , and you want to find when the temperature equals exactly 35°C. This means you need to substitute 35 for $T$ in the equation, giving you $35 = 80 \cdot (0.95)^t + 20$ . Rearranging this gives the equivalent form $80 \cdot (0.95)^t + 20 = 35$ , which is answer choice C. Let's see why the other options are incorrect. Choice A ( $80 \cdot (0.95)^t + 35 = 20$ ) incorrectly adds 35 to the exponential term instead of setting the entire expression equal to 35. Choice B ( $80 \cdot (0.95)^t = 35$ ) forgets to include the constant term 20 from the original equation, which represents the ambient temperature the object approaches. Choice D ( $35 = 80 \cdot (0.95)^t - 20$ ) changes the sign of the constant term from +20 to -20, which would represent a completely different cooling scenario. Remember this key strategy: when solving "when does [expression] equal [value]" problems, substitute the target value for the variable in the original equation. Don't modify the structure of the given formula—just set it equal to your target and solve from there.

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