Page 34 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

Example 25

If x . 0, find the range of values of x which satisfy each of the following inequalities. 1
(a) 5x – 2 , 3 x + 5 (b) 3x  2 + 1
2 8 x
3
Solution: (a) 5x – 2 , x + 5
2 8
5
5x – 3 x , + 2
2 8
7 x , 21
2 8
x , 3
4
Since x . 0, the range of values of x for 5x – 2 , 3 x + 5 is 0 , x , 3 .
2 8 4
(b) 3x  2 + 1
x
3x – 2 – 1  0
x
1
Since x . 0, 3x – 2 – 1 2 · x  0
x
i.e. 3x – 2x – 1  0
2
(3x + 1)(x – 1)  0
Since x . 0, (3x + 1) is always positive.
Hence, x – 1  0
or x  1
The range of values of x for the inequality
3x  2 + 1 is x  1.
x

Example 26

Show that the following inequalities are true for all x  R.
(a) 2x + 8x + 9 . 0 (b) –3x + 2x – 5 , 0
2
2
2
Solution: (a) Let h(x) = 2x + 8x + 9
1
2
= 2 x + 4x + 9 2
2
1
2
2
2
= 2 x + 4x + 2 – 2 + 9 2
2
3
2
= 2 (x + 2) + 1 4
2
= 2(x + 2) + 1
2
2
Now, for all x  R, (x + 2)  0
Hence, 2x + 8x + 9 . 0 for all x  R.
2
2
(b) Let k(x) = –3x + 2x – 5
1
2
= –3 x – 2 x + 5 3 2
3
1
3
1
2
= –3 x – 2 x + – 1 2 2 – – 1 2 2 + 5 4
3
3
3
3
31
= –3 x – 1 2 2 + 14 4
9
3
1
= –3 x – 1 2 2 – 14
3
3
31
01a STPM Math T T1.indd 31 3/28/18 4:20 PM