Moderate -0.5 This is a straightforward exponential inequality requiring students to express both sides as powers of 2 and solve a linear inequality. While it involves multiple steps (rewriting 64 as 2^6, simplifying the exponent, solving 3x+1<6), these are routine techniques for P3 level with no conceptual challenges beyond standard index manipulation.
Use law of the logarithm of a product, quotient or power
M1
Obtain a correct linear inequality in any form, e.g. \(\ln 3 + (3x+1)\ln 2 < \ln 8\)
A1
Accept, for example, \(2 - \frac{\ln 3}{3\ln 2}\)
Obtain final answer \(x < \frac{\ln\frac{4}{3}}{\ln 8}\), or equivalent
A1
Total
3
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use law of the logarithm of a product, quotient or power | M1 | |
| Obtain a correct linear inequality in any form, e.g. $\ln 3 + (3x+1)\ln 2 < \ln 8$ | A1 | Accept, for example, $2 - \frac{\ln 3}{3\ln 2}$ |
| Obtain final answer $x < \frac{\ln\frac{4}{3}}{\ln 8}$, or equivalent | A1 | |
| **Total** | **3** | |
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1 Find the set of values of $x$ for which $3 \left( 2 ^ { 3 x + 1 } \right) < 8$. Give your answer in a simplified exact form.\\
\hfill \mbox{\textit{CAIE P3 2020 Q1 [3]}}