Fold the rational graph above the axis

(i) Find the exact roots of ∣(4x−5)/(x−2)∣=3x|(4x-5)/(x-2)|=3x. marks

(ii) Sketch that absolute-value curve and y=3x together. Hence solve ∣(5−4x)/(3x−6)∣>x|(5-4x)/(3x-6)|>x exactly. marks

[2020/YIJC/Prelim/I/4]

Total marks: 8

Solution

(i)

Absolute-value equations

Find the exact roots of ∣4x−5x−2∣=3x\left|\dfrac{4x-5}{x-2}\right|=3x.

Hint

  1. The right side 3x3x must be ≥0\ge0, so x≥0x\ge0.

  2. Split: 4x−5x−2=3x\frac{4x-5}{x-2}=3x or 4x−5x−2=−3x\frac{4x-5}{x-2}=-3x.

  3. Solve each quadratic and reject any negative root.

Answer · (i)

x=(5−10)/3, 5/3, (5+10)/3x=(5-\sqrt{10})/3,\ 5/3,\ (5+\sqrt{10})/3.

(ii)a

Sketching a modulus curve

Sketch the graph of y=∣4x−5x−2∣y=\left|\dfrac{4x-5}{x-2}\right| and the line y=3xy=3x on the same diagram, labelling asymptotes and intercepts.

Explore this idea

Hint

  1. Sketch y=4x−5x−2=4+3x−2y=\frac{4x-5}{x-2}=4+\frac3{x-2} first: asymptotes x=2x=2 and y=4y=4.

  2. Find its intercepts, then reflect the part below the xx-axis upwards.

  3. Add y=3xy=3x and mark the three intersections from (i).

Absolute rational curve and line
Absolute rational curve and lineScroll across to see the full diagram.

Answer · (ii)a

Asymptotes x=2,y=4x=2,y=4; intercepts (0,5/2),(5/4,0)(0,5/2),(5/4,0); the three intersections above.

Explore the idea

The folded curve and the line y = 3x

Switch on the features.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Folded rational curve
  • Y = 3x
  • Asymptotes
  • Three intersections
Ideas to try
Find the sharp corner on the xx-axis. Why is it sharp?
(ii)b

Inequalities from graphs

Hence solve ∣5−4x3x−6∣>x\left|\dfrac{5-4x}{3x-6}\right|>x exactly.

Explore this idea

Hint

  1. Rewrite to match the sketch: ∣5−4x3x−6∣=13∣4x−5x−2∣\left|\frac{5-4x}{3x-6}\right|=\frac13\left|\frac{4x-5}{x-2}\right|.

  2. So the inequality says ∣4x−5x−2∣>3x\left|\frac{4x-5}{x-2}\right|>3x.

  3. Read the intervals where the folded curve is above the line.

Answer · (ii)b

x<(5−10)/3x<(5-\sqrt{10})/3 or 5/3<x<25/3<x<2 or 2<x<(5+10)/32<x<(5+\sqrt{10})/3.

Explore the idea

Where is the folded curve above the line?

The vertical guides mark the three intersections and the asymptote.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Folded curve
  • Y = 3x
  • Boundary values
  • Asymptote x = 2
Ideas to try
Test x=1x=1: the folded curve is 1 and the line is 3. Is the curve above the line? Which interval does x=1x=1 belong to?