Hint
The base is positive and not equal to one. Equate the two exponents.
Worked solution
(a)
Compare p(x−h) with k, reversing the direction exactly when 0<a<1; then divide by positive p.
Answer: \(x > 12\)
(b)
Use monotonicity of the exponential.
The base increases with its exponent; compare exponents accordingly.
Answer: \(x > 12\)
(c)
Solve \(\displaystyle p(x-h)=k+q\).
\(\displaystyle \frac{27}{2}\)
Answer: \(\displaystyle \frac{27}{2}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Use whether the base is above or below one to choose the correct inequality direction.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A base between zero and one reverses the inequality.
(b) · 2 marks
- 1 mark: Use monotonicity of the exponential.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not reverse an inequality twice without a reason.
(c) · 2 marks
- 1 mark: Solve \(\displaystyle p(x-h)=k+q\).
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A base between zero and one reverses the inequality.
Total: 6 marks.