FP2 June 2007 Q5
5. Find the set of values of \(x\) for which \[\frac{x + 1}{2x - 3} \lt \frac{1}{x - 3}\]
| Scheme | Marks |
|---|---|
| \(1\tfrac{1}{2}\) and 3 are ‘critical values’, e.g. used in solution, or both seen as asymptotes. | B1 |
| \((x + 1)(x - 3) = 2x - 3 \Rightarrow x(x - 4) = 0\) \(x = 4,\ x = 0\) M1: Attempt to find at least one other critical value | M1A1, A1 |
| \(0 \lt x \lt 1\tfrac{1}{2},\ 3 \lt x \lt 4\) M1: An inequality using \(1\tfrac{1}{2}\) or 3 | M1A1, A1 |
| (7 marks) |
Notes
First M mark can be implied by the two correct values, but otherwise a method must be seen. (The method may be graphical, but either \((x =)\) 4 or \((x =)\) 0 needs to be clearly written or used in this case).
Ignore ‘extra values’ which might arise through ‘squaring both sides’ methods.
\(\leqslant\) appearing: maximum one A mark penalty (final mark).