Fold the rational graph above the axis · source reference

Complete question

(i) Find the exact roots of $|(4x-5)/(x-2)|=3x$. [3] (ii) Sketch that absolute-value curve and y=3x together. Hence solve $|(5-4x)/(3x-6)|>x$ exactly. [5]

[2020/YIJC/Prelim/I/4]

Printed working

Part (i)

$$\left|\frac{4x-5}{x-2}\right|=3x\quad\Longrightarrow\quad \frac{4x-5}{x-2}=3x\quad\text{or}\quad\frac{4x-5}{x-2}=-3x.$$

First equation:

$$\begin{aligned} 4x-5&=3x^2-6x\\ 3x^2-10x+5&=0\\ x&=\frac{10\pm\sqrt{10^2-4(3)(5)}}6 =\frac{10\pm\sqrt{40}}6=\frac53\pm\frac{\sqrt{10}}3. \end{aligned}$$

Second equation:

$$4x-5=-3x^2+6x\Rightarrow3x^2-2x-5=0\Rightarrow(3x-5)(x+1)=0.$$

This gives $x=5/3$ or $-1$; reject $-1$ since $x\ge0$. Hence the roots are $x=5/3$ or $5/3\pm\sqrt{10}/3$.

Part (ii)

The underlying rational curve is $\dfrac{4x-5}{x-2}=4+\dfrac3{x-2}$. Reflect its negative-y portion in the x-axis to obtain $y=\left|(4x-5)/(x-2)\right|$.

The source sketch labels $y=4$, $x=2$, $(0,5/2)$, $(5/4,0)$, the three intersections $x=5/3-\sqrt{10}/3$, $5/3$, $5/3+\sqrt{10}/3$, and $y=3x$.

$$\begin{aligned} \left|\frac{5-4x}{3x-6}\right|&>x\\ \left|\frac{(-1)(4x-5)}{3(x-2)}\right|&>x\\ \left|\frac{4x-5}{x-2}\right|&>3x. \end{aligned}$$

From the graphs, $x<5/3-\sqrt{10}/3$ or $5/3<x<2$ or $2<x<5/3+\sqrt{10}/3$.

Teacher margin notes

Note: $x\ge0$, as $\left|(4x-5)/(x-2)\right|\ge0$.

To sketch the graph of $y=|f(x)|$: obtain it by reflecting the part of the curve $y=f(x)$ where $y<0$ above the x-axis, while the part where $y\ge0$ remains unchanged.

Source diagrams

Provenance

Question page references: 2. Solution page references: 13, 14. The original compilation header and printed page numbers are preserved in the edition notes. Original PDFs and page images remain only in the private QA archive.