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Solve for xx.\newline (x - 5)(x + 5) > 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−5)(x+5)>0 (x - 5)(x + 5) > 0 \newlineWrite a compound inequality like 1<x<31 < x < 3 or like x<1x < 1 or x>3x > 3.______
  1. Divide Intervals: Divide the number line into intervals using the critical points.\newlineIntervals: (−∞,−5)(-\infty, -5), (−5,5)(-5, 5), (5,∞)(5, \infty).
  2. Test (−∞,−5)(-\infty, -5): Test the interval (−∞,−5)(-\infty, -5) to determine the sign of (x−5)(x+5)(x - 5)(x + 5). Choose x=−6x = -6: (−6−5)(−6+5)=(−11)(−1)=11(-6 - 5)(-6 + 5) = (-11)(-1) = 11, which is positive.
  3. Test (−5,5) (-5, 5) : Test the interval (−5,5) (-5, 5) to determine the sign of (x−5)(x+5) (x - 5)(x + 5) .\newlineChoose x=0 x = 0 : (0−5)(0+5)=(−5)(5)=−25 (0 - 5)(0 + 5) = (-5)(5) = -25 , which is negative.
  4. Test \(5, \infty): Test the interval \(5, \infty) to determine the sign of x - \(5)(x + 55). Choose x = \(6): \(6 - 55)(66 + 55) = (11)(1111) = 1111, which is positive.
  5. Find Solution: Since we want (x - 5)(x + 5) > 0, we look for intervals where the product is positive.\newlineThe solution is x < -5 or x > 5.

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