Can you find domain?
If the domain of the logarithm lo g − x ( x 2 − 5 x − 2 4 − x − 2 ) is ( − ∞ , a ] , find a .
The answer is -3.
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1 solution
How did you solve the second restriction?
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Squaring, x^2-5x-24> x^2+4x+4=>-9x> 28=> x <-28/9. Solving with first restriction, a=-3
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That's a mistake. You can't square the inequality because we don't how's the absolute value. For example:
4 > − 5 , but clearly 1 6 ≯ 2 5
Even worse: x ≥ 0 , so x ≥ 0 for all x ∈ ℜ ?
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@Mateo Matijasevick – We know that for x=-28/9 both sides are equal. For x <-28/9, rhs is negative while lhs is positive. Thus lhs> rhs. Lhs will again be positive at x> 8 but rhs will be more positive. Hence verified . And one more thing for x 1 / 2 >=0, x>=0 since negative number's root is not real as you mentioned in your comment!
The first restriction is square root which gives ( − ∞ , − 3 ) U ( 8 , ∞ ) . Second restriction is log, in which x 2 − 5 x − 2 4 > x + 2 . Squaring, x 2 − 5 x − 2 4 > x 2 + 4 x + 4 = > − 9 x > 2 8 = > x < − 2 8 / 9 . Solving with first restriction,Final answer is ( − ∞ , − 3 ) . Thus a = − 3 . Verification:We know that for x = − 2 8 / 9 both sides are equal. For x < − 2 8 / 9 , rhs is negative while lhs is positive. Thus lhs> rhs. Lhs will again be positive at x > 8 but rhs will be more positive. And obviously base of log will be negative for x > 8