Inequalities: Rational inequality & \(\mathrm{e}^x\) — EJC 2025 H2 Math Prelim Paper 1
Eunoia Junior College2025 PrelimPaper 1●○○ Introductory5 marks
What this question tests
Rational inequality & \(\mathrm{e}^x\).
Question
- Solve the inequality \[ x + 1 > \frac{1}{x-1}, \] giving your answer in exact form.
- Hence solve, in exact form, \[ \mathrm{e}^x + 1 > \frac{1}{\mathrm{e}^x - 1}. \]
Show full worked solution▾
\[\begin{aligned}
x + 1 - \frac{1}{x-1} &> 0\\[4pt]
\frac{(x+1)(x-1) - 1}{x-1} &> 0\\[4pt]
\frac{x^2 - 1 - 1}{x-1} &> 0\\[4pt]
\frac{x^2 - 2}{x-1} &> 0\\[4pt]
\frac{(x+\sqrt{2})(x-\sqrt{2})}{x-1} &> 0
\end{aligned}\]
Critical values: \(x = -\sqrt{2}\), \(x = 1\), \(x = \sqrt{2}\).
Answer: (a): \(-\sqrt{2} < x < 1\) or \(x > \sqrt{2}\) (b): \(x < 0\) or \(x > \tfrac{1}{2}\ln 2\)