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Linear Inequalities in One or Two Variables Flashcards

6 cards from real Bluebook SAT Test practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

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  1. Which value of x does NOT satisfy the inequality [3(x − 4)]/2 − [(5 − x)]/3 > x − 1?

    Answer: x = 8

    Multiply all terms by 6 to clear fractions: 9(x − 4) − 2(5 − x) > 6(x − 1), which simplifies to 9x − 36 − 10 + 2x > 6x − 6, then 11x − 46 > 6x − 6, giving 5x > 40, so x > 8 (strict). Since x = 8 makes the inequality 40 > 40, which is false, it does NOT satisfy the inequality. All other values (9, 10, 15) are strictly greater than 8 and do satisfy it.

  2. How many ordered pairs (x, y) satisfy BOTH inequalities y > 2x + 5 and y < 2x − 3 simultaneously?

    Answer: None

    Both boundary lines y = 2x + 5 and y = 2x − 3 are parallel (same slope of 2). The first inequality requires y to lie strictly above 2x + 5, while the second requires y to lie strictly below 2x − 3. Since 2x + 5 > 2x − 3 for all x (because 5 > −3), the line y = 2x + 5 is always above y = 2x − 3. It is impossible to be simultaneously above the higher line and below the lower line. The system has no solution.

  3. Which of the following is equivalent to −5 ≤ 7 − 3x ≤ 13?

    Answer: −2 ≤ x ≤ 4

    Subtract 7 from all three parts: −12 ≤ −3x ≤ 6. Then divide by −3 — critically, dividing by a negative number reverses BOTH inequality signs: 4 ≥ x ≥ −2, which is written as −2 ≤ x ≤ 4. Choice A results from forgetting to flip the inequalities. Choice C confuses 'and' with 'or'. Choice D incorrectly converts the non-strict inequalities (≤) to strict ones.

  4. A theater charges $12 for adult tickets and $8 for student tickets. For a given performance, the theater must earn at least $960 in revenue and can sell no more than 100 total tickets. Which ordered pair (adults, students) satisfies ALL constraints, including non-negativity?

    Answer: (40, 60)

    The constraints are: 12x + 8y ≥ 960 (revenue), x + y ≤ 100 (capacity), x ≥ 0, y ≥ 0. Checking (40, 60): revenue = 12(40) + 8(60) = 480 + 480 = 960 ≥ 960 ✓; total tickets = 100 ≤ 100 ✓. Choice (35, 60) yields only $900 in revenue, failing the revenue constraint. Choices (45, 60) and (50, 55) both exceed 100 total tickets.

  5. Which ordered pair lies in the solution region of the system x + y > 4 AND x − y > 4?

    Answer: (5, 0)

    Test each point in both inequalities. (5, 0): x + y = 5 > 4 ✓ and x − y = 5 > 4 ✓ — satisfies both. (3, 2): x − y = 1, which is not > 4 ✗. (4, 1): x − y = 3, not > 4 ✗. (5, −1): x + y = 4, which fails the strict inequality 4 > 4 ✗. Adding the two original inequalities gives 2x > 8, so x > 4 is a necessary condition — this alone eliminates (3, 2) and (4, 1).

  6. A set of values is described by the compound inequality −2 ≤ x ≤ 5. Which of the following describes all values of x that are NOT in this set?

    Answer: x 5

    The set −2 ≤ x ≤ 5 can be written as (x ≥ −2) AND (x ≤ 5). By De Morgan's Law, the negation of (P AND Q) is (NOT P) OR (NOT Q). So: NOT(x ≥ −2) OR NOT(x ≤ 5) = (x 5). The boundaries −2 and 5 are IN the original set, so they must NOT be in the complement — ruling out Choice B, which uses ≤ and ≥ instead of strict inequalities. Choice D describes an empty set (no number can be simultaneously less than −2 and greater than 5).