Syllabus 9740 - Inequalities

Chow Kim Wan

1. Without using a calculator, solve the inequalityx − 1

2x + 16≥

x − 13x − 1

.

2. (a) In a farm, there were 3 different types of animals: chickens, horses and sheep. The farmer confirmed that the number ofsheep is twice the number of chickens. He also counted a total of 1250 animal legs. Due to his handwriting, he was notsure if there were 250 animals or 350 animals in total.Find the correct number of chickens, horses and sheep.

(b) By using an algebraic method, solve the inequality2x − 12x + 11

> x + 1.

Hence, solve the inequalityln x2 − 1ln x2 + 11

> ln x + 1.

3. (a) The equation of a curve is given by y = a√

x +bx+ cx2 + d, where a, b, c and d are constants. It is given that the curve

passes through the points (1,−9),(4,

234

)and

(9,

2173

). It is also known that the curve has a stationary point at (1,−9).

Determine the equation of the curve.

(b) i. Prove that94

x2 − 3x +52is positive for all real values of x.

ii. Solve9x3 − 12x2 + 10x4x2 − 24x + 32

< 0.

iii. Hence, solve9(cos−1 x

)3− 12

(cos−1 x

)2+ 10

(cos−1 x

)4(cos−1 x

)2− 24

(cos−1 x

)+ 32

< 0, leaving your answer in exact form.

4. Without using a calculator, solve the inequality9

20 + 9x + x2 ≥ −4.

5. (a) Using an algebraic method, solve the inequality2x + 39x2 + 3

≤ 5.

(b) Hence, by completing the square or otherwise, solve the inequality

2x + 25x2 − 14x + 52

> 5.

6. Without the use of a calculator, solve the inequality6x

x2 − 8x + 24> 1.

Hence, find the range of values of x that satisfy6ex

3e2x − 8ex + 8> 1.

1