Equations, inequalities and graphs

On the axes, sketch the graph of $y = (1-x)(x-5)(2x-5)$, stating the intercepts with the axes.

Hence solve the inequality $(1-x)(x-5)(2x-5) \leq 0$.

On the axes, sketch the graph of $y = -\frac{1}{5}(x+2)(2x-1)(x+5)$, stating the intercepts with the axes.

Hence solve the inequality $-\frac{1}{5}(x+2)(2x-1)(x+5) \geq 0$.

The diagram shows the graph of $y = |\mathrm{f}(x)|$, where $\mathrm{f}(x)$ is a cubic polynomial. Find, in factorised form, the possible expressions for $\mathrm{f}(x)$.

Solve the inequality $|5x - 2| \leq |4x + 1|$.

On the axes, sketch the graphs of $y = 2x + 5$ and $y = |4x - 3|$, stating the intercepts with the coordinate axes.

Solve the inequality $|4x - 3| < 2x + 5$.

The diagram shows the graph of $y = |\mathrm{f}(x)|$, where $\mathrm{f}(x)$ is a cubic. Find the possible expressions for $\mathrm{f}(x)$.

On the axes below, sketch the graph of $y = |2x+1|$ and the graph of $y = |4(x-1)|$, stating the coordinates of the points where the graphs meet the coordinate axes.

Find the exact solutions of the equation $|2x+1| = |4(x-1)|$.

On the axes, draw the graph of $y = |3x^2 + 13x - 10|$, stating the coordinates of the points where the graph meets the axes.

Find the set of values of the constant $k$ such that the equation $k = |3x^2 + 13x - 10|$ has exactly 2 distinct roots.