Worked Examples: SAT Strategy
Example 1: Easy (Translating Constraints)
Question: For a snowstorm in a certain town, the minimum rate of snowfall recorded was $0.6$ inches per hour, and the maximum rate was $1.8$ inches per hour. Which inequality represents the rate of snowfall $s$?
SAT Strategy: Look for key boundary words like "minimum", "maximum", "at least", or "at most".
- "Minimum was 0.6" means $s$ must be greater than or equal to $0.6$ ($0.6 \le s$).
- "Maximum was 1.8" means $s$ must be less than or equal to $1.8$ ($s \le 1.8$).
- Combine them into a single compound inequality.
- Answer: $0.6 \le s \le 1.8$.
Example 2: Medium (Solving and Flipping the Sign)
Question: What is the greatest possible integer solution to the inequality $-3x + 12 \ge 24$?
SAT Strategy: Treat the inequality exactly like an equation, with one major exception:
if you multiply or divide by a negative number, you must flip the inequality sign.
- Subtract 12 from both sides: $-3x \ge 12$.
- Divide by $-3$. Because we are dividing by a negative, flip the $\ge$ to $\le$.
- $x \le -4$.
- The question asks for the greatest integer that satisfies this. The integers less than or equal to $-4$ are $-4, -5, -6, \dots$. The greatest is $-4$.
- Answer: $-4$.
Example 3: Hard (Systems of Inequalities)
Question: The point $(8, 2)$ in the $xy$-plane is a solution to which of the following systems of inequalities?
A) $x > 0, y < 0$ B) $x < 0, y > 0$ C) $x > 0, y > 0$ D) $x < 0, y < 0$
SAT Strategy: You do not need to graph these! The phrase "is a solution to" simply means "makes the inequality true when you plug in the numbers."
- The point is $(8, 2)$, meaning $x = 8$ and $y = 2$.
- Test $x = 8$: $8 > 0$ is True. $8 < 0$ is False. This eliminates B and D.
- Test $y = 2$: $2 > 0$ is True. $2 < 0$ is False. This eliminates A.
- Answer: C.
Practice Questions
Level 1: Easy (Foundational SAT)
- $x - 7 \ge 12$. What is the least possible value of $x$? (Grid-in)
- A delivery truck can carry a maximum load of $2,000$ pounds. If a worker is loading boxes that weigh $50$ pounds each, which inequality represents the number of boxes, $b$, the truck can carry?
- A) $50b \ge 2000$
- B) $50b \le 2000$
- C) $50 + b \le 2000$
- D) $b \le 50(2000)$
- $3x + 15 \le 45$. What is the greatest possible value of $x$? (Grid-in)
- Which of the following points $(x, y)$ is a solution to the inequality $y > x + 2$?
- A) $(0, 0)$
- B) $(2, 4)$
- C) $(3, 6)$
- D) $(5, 1)$
- Ty set a goal to walk at least 24 kilometers every day. If Ty walks at an average speed of 4 kilometers per hour, what is the minimum number of hours Ty must walk to fulfill the goal? (Grid-in)
Level 2: Medium (Standard SAT)
- $-2x + 5 \ge 17$. Which of the following represents all solutions to the given inequality?
- A) $x \le -6$
- B) $x \ge -6$
- C) $x \le 6$
- D) $x \ge 6$
- $2(x - 4) < 3x - 10$. What is the least integer value of $x$ that satisfies the inequality? (Grid-in)
- A student is buying pens and notebooks. Pens cost $\$2$ each and notebooks cost $\$5$ each. The student wants to buy exactly $12$ items and has a maximum budget of $\$35$. Which system of inequalities represents this situation, where $p$ is the number of pens and $n$ is the number of notebooks?
- A) $p + n \ge 12$ and $2p + 5n \le 35$
- B) $p + n = 12$ and $2p + 5n \le 35$
- C) $p + n = 12$ and $2p + 5n \ge 35$
- D) $p + n \le 12$ and $2p + 5n = 35$
- Which point is a solution to the system of inequalities $y < 2x - 3$ and $y > -x + 1$?
- A) $(0, 0)$
- B) $(1, -1)$
- C) $(4, 0)$
- D) $(4, 6)$
- $y \ge 4x + 8$. For which of the following tables are all the values of $x$ and their corresponding values of $y$ solutions to the given inequality?
- A) $(2, 19), (4, 30), (6, 41)$
- B) $(2, 8), (4, 16), (6, 24)$
- C) $(2, 15), (4, 25), (6, 30)$
- D) $(2, 16), (4, 24), (6, 32)$
Level 3: Hard (Advanced SAT)
- $y \le -3x + 15$
$x \ge 4$
If $(x, y)$ is a solution to the system of inequalities above, what is the maximum possible value of $y$? (Grid-in) - A business owner budgets $\$100$ to purchase supplies. Small boxes cost $\$2$ each, and large boxes cost $\$5$ each. The owner must purchase a minimum of $30$ boxes total. What is the maximum number of large boxes the owner can purchase? (Grid-in)
- $ax + 5 < 17$. In the given inequality, $a$ is a positive constant. If $x = 3$ is a solution to the inequality, but $x = 4$ is NOT a solution, which of the following could be the value of $a$?
- A) $2$
- B) $3.5$
- C) $4$
- D) $5$
- $|x - 2| < 5$. How many integer values of $x$ satisfy the given absolute value inequality? (Grid-in)
- $y > 13x - 18$. For which of the following tables are all the values of $x$ and their corresponding values of $y$ solutions to the given inequality?
- A) $(3, 21), (5, 47), (8, 86)$
- B) $(3, 26), (5, 42), (8, 86)$
- C) $(3, 16), (5, 42), (8, 81)$
- D) $(3, 26), (5, 52), (8, 91)$
Level 4: Very Hard (Slightly beyond SAT)
- $y \le 5x + c$. In the given inequality, $c$ is a constant. If the point $(3, 10)$ is a solution to the inequality, what is the minimum possible value of $c$? (Grid-in)
- $\frac{x}{3} - \frac{x}{4} > 1$. What is the least integer value of $x$ that satisfies the inequality? (Grid-in)
- $y \ge \frac{1}{2}x + 3$
$y \le \frac{1}{2}x - 2$
How many solutions $(x, y)$ exist for the given system of inequalities in the $xy$-plane?
- A) Zero
- B) Exactly one
- C) Exactly two
- D) Infinitely many
- $y \le -x + 4$
$y \ge x$
$y \ge 0$
The solution to the given system of inequalities forms a triangular region in the $xy$-plane. What is the area of this region? (Grid-in) - $y \le -2x + 20$
$y \ge 3x - 15$
If $(x, y)$ is a solution to the system of inequalities above, what is the maximum possible value of $x$? (Grid-in)
🔒 Answer Key & Brief Explanations
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