Standard +0.8 This requires manipulating logarithm laws to combine logs, converting the inequality to exponential form, then solving a rational inequality with careful domain restrictions (ensuring arguments are positive). The multiple algebraic steps, factorization, and sign analysis of the rational expression make this moderately challenging, though the techniques are all standard A-level material.
Solve: if \(x > 1\) then \(x < \frac{3}{2}\); if \(x < 1\), then \(x > \frac{2}{3}\)
M1
The solution set is \(1 < x < \frac{3}{2}\)
A1
[5]
*Note: Equivalent marks for candidates who do not divide out the algebraic fraction and attempt to solve a quadratic inequality.*
Correctly combine the logs $\log_3\left(\frac{2x^2 - x}{2x^2 - 3x + 1}\right) > 1$ | M1 |
Simplify $\log_3\frac{x}{x-1} > 1$ | A1 |
Rewrite as $\frac{x}{x-1} > 3$ | M1 |
Solve: if $x > 1$ then $x < \frac{3}{2}$; if $x < 1$, then $x > \frac{2}{3}$ | M1 |
The solution set is $1 < x < \frac{3}{2}$ | A1 | [5]
*Note: Equivalent marks for candidates who do not divide out the algebraic fraction and attempt to solve a quadratic inequality.*