You solve x² - x - 6 > 0, find roots x = -2 and x = 3, and the parabola opens upward. What is the solution set?
A(-2, 3)
B(-∞, -2) ∪ (3, ∞)
C(-∞, -2)
D(3, ∞)
When a parabola opens upward, it is BELOW the x-axis between the roots and ABOVE it outside them. So x² - x - 6 > 0 (positive) gives the two outer regions: x < -2 or x > 3. The tempting wrong answer (-2, 3) is actually the solution to x² - x - 6 < 0. Recognizing which region satisfies the inequality requires understanding the parabola's shape, not just finding the roots.
Question 2 Multiple Choice
A student solves (x - 2)(x - 3) < 0, finds roots 2 and 3, and writes the solution as 'x < 2 or x > 3.' What error did they make?
AThey factored incorrectly — the roots should be -2 and -3
BFor < 0 with an upward parabola, the solution is the interval BETWEEN the roots: (2, 3), not outside them
CThey should have included the endpoints: x ≤ 2 or x ≥ 3
DQuadratic inequalities cannot be solved by factoring
The sign pattern of an upward parabola is: positive | negative | positive across the three intervals defined by its roots. The expression (x - 2)(x - 3) is negative only between the roots, giving the bounded interval (2, 3). Writing 'x < 2 or x > 3' confuses the solution to > 0 with the solution to < 0 — the two solutions are exact complements.
Question 3 True / False
The solution to x² - 4 > 0 can be written as the single interval x > 2.
TTrue
FFalse
Answer: False
x² - 4 = (x + 2)(x - 2) has roots at x = -2 and x = 2. Since the parabola opens upward, it is positive in TWO regions: x < -2 and x > 2. Writing only 'x > 2' misses the entire left branch. The correct solution is (-∞, -2) ∪ (2, ∞).
Question 4 True / False
You cannot solve a quadratic inequality by treating it like a linear inequality — algebraically isolating x on one side — because the parabola's sign changes in a way that linear manipulation cannot track.
TTrue
FFalse
Answer: True
Linear inequalities have a consistent direction of solution (x > k or x < k). Quadratic inequalities produce sign patterns that depend on the parabola's shape, and solutions are often unions of intervals. Additionally, dividing by a term containing x is illegal since you don't know its sign. The correct method is always: find zeros first, then test signs in each resulting interval.
Question 5 Short Answer
Why do solutions to quadratic inequalities often consist of two separate intervals rather than one connected interval, and how does the parabola's graph make this clear?
Think about your answer, then reveal below.
Model answer: An upward-opening parabola dips below the x-axis only between its roots and rises above it on both outer sides. So a > 0 inequality picks up both outer regions (a union of two unbounded intervals), while a < 0 inequality picks up the single bounded region between the roots. The graph makes this visual: you can see exactly where the curve lies above or below the axis, and the solution is simply the x-projection of those regions.
This is why quadratic solutions don't look like linear ones. Linear inequalities produce a half-line; quadratic inequalities produce a bounded segment or a union of two rays, depending on direction. Connecting the algebraic sign analysis to the geometric parabola is the key insight that makes the general method intuitive.