Page 39 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

Exercise 1.6

1 1. With the aid of a sketch graph, solve each of the following inequalities.
(a) (x + 1)(x – 2) < 0 (b) (x – 3)(x + 5) . 0
(c) (2x – 3)(x + 4) , 0 (d) (2x + 1)(4x – 1)  0
1
(e) ( x + 5)(x – 3) < 0 (f) (x – 2)(5x + 2) . 0
2
2. Solve each of the following inequalities.
2
(a) x  9 (b) x + 2x + 1 . 0
2
(c) x(x + 1) < –2(2x + 3) (d) 5x < 3x + 2
2
2
2
2
2
(e) (x – 2) . 9x (f) 3x – 2x  x + 3x + 3
3. Find the range of values of x such that each of the following inequalities is valid.
(a) (x + 2)(x – 1)(x + 3) , 0 (b) (x – 2) (x + 1) < 0
2
(c) x + 3x – 4  0 (d) 2x + 3x – 3x , 2
3
2
3
2
3
2
2
2
(e) x(5x + 8) < 1 (47x – 48) (f) 2x  7x + 17x – 10
2
4. Find the range of values of x which satisfy each of the following inequalities.
(a) 4 . 2 – x (b) 4 – 5x . 3
x + 3 1 – 2x
(c) 14  2x – 1 (d) 13 – 4x , 35
x – 2 x – 1 x – 3
(e) 9 < 7x + 5 (f) x + 1 . 3
4 – x x + 3 2x – 1 x – 2
5. Find the set of values of x which satisfy each of the following inequalities.
(a) |x – 2| , 1 (b) |x – 3|  5 (c) |3x + 4| . 5
(d) |2x – 5| < 11 (e) |x|  |x – 1| (f) 2|x – 2| , |x – 3|
(g) 3|x + 2| < |x – 6| (h) 5|2x – 3| . 4|x – 5| (i) |2x + 1| , 3x + 2
2
(j)  x  , 2 (k)  x – 4  < 3 (l)  x + 1  , 1
x + 4 x x – 1
Partial fractions
2
Let f(x) = 2x + 1 and g(x) = x + 3x + 2. When f(x) is divided by g(x), the resulting function
2x + 1 2x + 1
–––––––––– ≡ ––––––––––––
2
x + 3x + 2 (x + 1)(x + 2)
is known as a rational function.
Notice that in the above case, f(x) is linear and g(x) is quadratic, i.e. the degree of f(x) is less than
the degree of g(x). If f(x) is a polynomial function of degree m, and g(x) is a polynomial of degree n,
f(x)
where m , n, then h(x) = — is considered a proper rational function.
g(x)
x + 1
2

2
x + 2

For example, –––––––––––– , –––––––––– and –––––– are all proper rational functions.
(x – 1)(x + 3) 2x + x + 1 3x + 1
3
f(x)
However, if m  n, then h(x) = — is considered an improper rational function.
g(x)
2
2
For example, ––––––––––– , ––––––––––– and ––––––––––––––– are all improper rational functions.
x + 2x + 3 x + 3x + 1
2x + x – 3x + 1
3
2
2
x + 1 x + 4x – 5 (x + 1)(x + 2)
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