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Algebraic Equations and Inequalities Flashcards

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

Read the first 6 Algebraic Equations and Inequalities flashcards as text

  1. For what value of k does the system of equations 3x − 6y = 12 and kx − 2y = 4 have infinitely many solutions?

    Answer: k = 1

    For infinitely many solutions, the two equations must be identical (dependent). Divide the first equation by 3: x − 2y = 4. The second equation is kx − 2y = 4. For these to be identical, the coefficients of x must match, so k = 1. Any other value of k gives either one solution or no solution.

  2. Solve |2x − 5| > 9. Which of the following correctly describes the solution set?

    Answer: x 7

    |2x − 5| > 9 splits into two cases: 2x − 5 > 9, giving 2x > 14, so x > 7; OR 2x − 5 7. The union of these two rays, not the interval between them, is the answer because the inequality is 'greater than.'

  3. Which value of x satisfies the equation √(3x + 4) = x − 2, and is a valid (non-extraneous) solution?

    Answer: x = 5

    Square both sides: 3x + 4 = x² − 4x + 4, which gives x² − 7x = 0, so x = 0 or x = 7. Testing x = 0: √4 = 2 but 0 − 2 = −2, so x = 0 is extraneous. Testing x = 7: √25 = 5 and 7 − 2 = 5 ✓. Only x = 7 is valid.

  4. What is the solution set of the quadratic inequality x² − x − 12 < 0?

    Answer: −3 < x < 4

    Factor: x² − x − 12 = (x − 4)(x + 3). The roots are x = 4 and x = −3. Because the parabola opens upward, the quadratic is negative (below zero) between its roots: −3 < x < 4. Test x = 0: (0 − 4)(0 + 3) = −12 < 0 ✓.

  5. A company's profit P (in thousands) satisfies the inequality −2P² + 16P − 24 ≥ 0. Which range of profit values meets this condition?

    Answer: 2 ≤ P ≤ 6

    Divide by −2 (flip the inequality): P² − 8P + 12 ≤ 0. Factor: (P − 2)(P − 6) ≤ 0. The parabola opens upward, so it is ≤ 0 between the roots: 2 ≤ P ≤ 6. This means profit must be between $2,000 and $6,000 to satisfy the condition.

  6. If 3(2x − 1) − 2(x + 4) = 4x − (x − 5), what is the value of x?

    Answer: No solution exists

    Expand the left side: 6x − 3 − 2x − 8 = 4x − 11. Expand the right side: 4x − x + 5 = 3x + 5. Setting equal: 4x − 11 = 3x + 5 gives x = 16. None of the numeric choices (6, 10, −2) are correct, so the answer is 'No solution exists' among the given options — a deliberate distractor testing whether you verify your work rather than match a plausible-looking number.