Basic Mathematics & Logarithm Question - None

Question

Solve the inequality: $ \dfrac{x(\pi^x - 7^x)(x-1)(x-3)}{(x-6)\log_{10}(x-4)} > 0 $. Find the solution set.

Accepted Answer

Step 1: Determine Domain

For log₁₀(x-4) to exist: x > 4

Also (x-6) ≠ 0, so x ≠ 6

Domain: (4, 6) ∪ (6, ∞)

Step 2: Analyze Each Factor

• x: Zero at x = 0 (outside domain), positive for x > 4

• (π^x - 7^x): Since π < 7, we have π^x < 7^x for all x. So this factor is always negative

• (x-1): Positive for x > 4

• (x-3): Positive for x > 4

• (x-6): Negative on (4,6), positive on (6,∞)

• log₁₀(x-4): Zero at x = 5; negative on (4,5); positive on (5,∞)

Step 3: Sign Table on Domain (4,6) ∪ (6,∞)

On (4, 5):

Numerator: (+)(−)(+)(+) = (−)

Denominator: (−)(−) = (+)

Quotient: (−)/(+) = (−) ✗

On (5, 6):

Numerator: (+)(−)(+)(+) = (−)

Denominator: (−)(+) = (−)

Quotient: (−)/(−) = (+) ✓

On (6, ∞):

Numerator: (+)(−)(+)(+) = (−)

Denominator: (+)(+) = (+)

Quotient: (−)/(+) = (−) ✗

∴ Answer: (5, 6)

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