Quick questions on Solving inequalities explained: H2 Mathematics Functions and Graphs

For higher-degree expressions, find all the roots, mark them on a number line, and determine the sign of the product in each interval (the sign flips at a simple root and stays the same across a repeated root of even multiplicity). Read off the intervals satisfying the inequality.

The critical trap: never cross-multiply by a denominator whose sign you do not know, because multiplying an inequality by a negative quantity reverses it. Two safe approaches:

The solution of $\mathrm{f}(x) > \mathrm{g}(x)$ is the set of $x$ where the graph of $\mathrm{f}$ lies above the graph of $\mathrm{g}$. Finding the intersection points and reading the regions is often the fastest route and is exactly how the graphing calculator helps.

$(x - a)(x - b) < 0$ holds between the roots; $> 0$ holds outside. Sketch to avoid reversing this.

At a double root the sign does not change; only odd-multiplicity roots flip the sign.

Solve $x^2 - 5x + 6 \geq 0$. [3 marks]

Solve $\dfrac{2}{x - 1} > 1$. [3 marks]

Explain why you should not multiply both sides of $\dfrac{1}{x} < 2$ by $x$. [2 marks]