Creating Equations: Creating and Solving Equations and Inequalities in One Variable (CCSS.A-CED.1)
Common Core High School Algebra · Learn by Concept
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Common Core High School Algebra › Creating Equations: Creating and Solving Equations and Inequalities in One Variable (CCSS.A-CED.1)
A streaming service charges a 10 dollar sign-up fee and 7 dollars per month. After $x$ months, the total cost is 52 dollars. Which equation best models the situation?
$10 + 7x = 52$
$7 + 10x = 52$
$10x + 7 = 52$
$10 + 7x = 42$
Explanation
Let $x$ be the number of months. The total cost is sign-up fee plus monthly rate times months: $10 + 7x = 52$. The choice $7 + 10x = 52$ swaps the fee and rate; $10x + 7 = 52$ treats the fee as the rate; $10 + 7x = 42$ is an arithmetic slip on the total.
Sam has 60 dollars saved and adds 15 dollars each week, aiming for at least 150 dollars. Let $w$ be the number of weeks. What is the solution to the inequality that models this scenario?
$w \le 6$
$w \ge 6$
$w \ge 5$
$w \ge 7$
Explanation
Model: $60 + 15w \ge 150$. Subtract 60 to get $15w \ge 90$, so $w \ge 6$. The choice $w \le 6$ reverses the inequality; $w \ge 5$ and $w \ge 7$ are arithmetic slips.
A rectangular garden has length $x$ meters and width $x - 3$ meters. Its area is 28 square meters. Which equation best models the situation?
$x^2 + 3x - 28 = 0$
$x^2 - 3x + 28 = 0$
$x^2 - 3x - 28 = 0$
$x^2 - 28x - 3 = 0$
Explanation
Area equals length times width: $x(x - 3) = 28 \Rightarrow x^2 - 3x - 28 = 0$. The option $x^2 + 3x - 28 = 0$ uses a wrong sign for width; $x^2 - 3x + 28 = 0$ moves 28 to the wrong side; $x^2 - 28x - 3 = 0$ mixes terms incorrectly.
A ride costs 4 dollars to start plus 3 dollars per mile. You can spend at most 25 dollars. Let $m$ be the number of miles. What is the solution to the inequality that models this scenario?
$m \ge 7$
$m \le 21$
$m \le 25$
$m \le 7$
Explanation
Model: $4 + 3m \le 25$. Subtract 4 to get $3m \le 21$, so $m \le 7$. The choice $m \ge 7$ reverses the direction; $m \le 21$ confuses dollars with miles; $m \le 25$ ignores the fixed fee and per-mile rate.
A tutoring service charges a 20 dollar monthly subscription plus 12 dollars per session. After attending $x$ sessions in one month, the total bill is 68 dollars. Which equation best models the situation?
$12 + 20x = 68$
$20 + 12x = 68$
$20x + 12 = 68$
$20 + 12x = 86$
Explanation
Let $x$ be the number of sessions. The total is subscription plus session fee times sessions: $20 + 12x = 68$. The options $12 + 20x = 68$ and $20x + 12 = 68$ swap the fee and rate; $20 + 12x = 86$ is an arithmetic slip on the total.
A music streaming plan charges a one-time setup fee of 12 dollars and 7 dollars per month. After $x$ months, the total paid is 61 dollars. Which equation best models the situation?
$12 + 7x = 61$
$7 + 12x = 61$
$12 + x = 61$
$7x = 61$
Explanation
Setup fee 12 is a constant and the monthly cost is $7$ per month, so total is $12 + 7x$. Setting this equal to 61 gives $12 + 7x = 61$. Choice B reverses the rate and fee, C drops the factor 7 on $x$, and D omits the setup fee.
You have 57 dollars to spend on movie tickets costing 9 dollars each, and you need at least 12 dollars left for snacks. What is the solution to the inequality that models this scenario?
$x \ge 5$
$x \le 6$
$x < 5$
$x \le 5$
Explanation
Spend on tickets plus snack money: $9x + 12 \le 57$. Then $9x \le 45$ so $x \le 5$. Choice B ignores the snack amount ($9x \le 57$), A reverses the inequality direction, and C uses a strict inequality instead of allowing $x=5$.
A ball's height in meters after $t$ seconds is $h(t) = -4t^2 + 8t + 2$. For how many seconds is the ball at least 6 meters high? What is the solution to the inequality that models this scenario?
$0 \le t \le 2$
$t = 0$ or $t = 2$
$t = 1$
$t \ge 1$
Explanation
Model with $-4t^2 + 8t + 2 \ge 6$. This simplifies to $-4t^2 + 8t - 4 \ge 0$, so dividing by $-4$ flips the inequality: $t^2 - 2t + 1 \le 0$, i.e., $(t-1)^2 \le 0$. This holds only at $t=1$. Other choices either give an interval, extraneous roots, or the wrong inequality direction.
A ride service charges a base fee of 3 dollars plus 2 dollars per mile. A trip costs 21 dollars. Which equation best models the situation for $x$ miles?
$3x + 2 = 21$
$2x + 3 = 21$
$2x = 21$
$3 + 2x = 21$
Explanation
Total cost is base fee plus per-mile cost: $3 + 2x = 21$. Choices A and B multiply the base fee or rate by $x$, and C omits the base fee entirely.
You have at most 90 minutes to practice. Your warm-up takes 15 minutes, and each drill takes 6 minutes. What is the solution to the inequality that models the number of drills $x$ you can complete?
$x \le 12$
$x \ge 12$
$x \le 13$
$15 + 6x \ge 90$
Explanation
Time limit model: $15 + 6x \le 90$. Then $6x \le 75$ so $x \le 12.5$, and with whole drills $x \le 12$. Choice B reverses the inequality, C rounds up incorrectly, and D uses the wrong inequality direction.