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Solve for xx.\newline (x - 3)(x - 6) < 0 \newlineWrite a compound inequality like 1 < x < 3 or like x < 1 or x > 3.

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Q. Solve for xx.\newline(x−3)(x−6)<0 (x - 3)(x - 6) < 0 \newlineWrite a compound inequality like 1<x<31 < x < 3 or like x<1x < 1 or x>3x > 3.
  1. Test Intervals: Test the intervals created by the critical points.\newlineIntervals: (−∞,3)(-\infty, 3), (3,6)(3, 6), (6,∞)(6, \infty).
  2. Check Sign: Check the sign of (x−3)(x−6)(x - 3)(x - 6) over (−∞,3)(-\infty, 3).\newlineChoose a test point, like x=0x = 0.\newlineSign of (0−3)(0 - 3): -.\newlineSign of (0−6)(0 - 6): -.\newlineSign of (x−3)(x−6)(x - 3)(x - 6): (−(-(-) = +.
  3. Check Sign: Check the sign of (x−3)(x−6)(x - 3)(x - 6) over (3,6)(3, 6).\newlineChoose a test point, like x=4x = 4.\newlineSign of (4−3)(4 - 3): +.\newlineSign of (4−6)(4 - 6): -.\newlineSign of (x−3)(x−6)(x - 3)(x - 6): (+)(−)=−(+)(-) = -.
  4. Check Sign: Check the sign of (x−3)(x−6)(x - 3)(x - 6) over (6,∞)(6, \infty). Choose a test point, like x=7x = 7. Sign of (7−3)(7 - 3): +. Sign of (7−6)(7 - 6): +. Sign of (x−3)(x−6)(x - 3)(x - 6): (+)(+)=+(+)(+) = +.
  5. Write Solution: (x - 3)(x - 6) < 0 is true for the interval (3,6)(3, 6).\newlineWrite the solution as a compound inequality.\newlineCompound inequality: 3 < x < 6.

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