082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

1.6 Solution of inequalities 33
(g)
(h)
(j)
(y +1)(y +2)(y +3)
(y +4) 3
(z +1) 10
(2z +1) 10 (i)
3k 2 +2k−1
k 3 +k 2 −4k+1
(q +1) 10
(q 2 +1) 6
3 Expresseachfraction in itssimplestform.
(a)
y 3 +2y
2y−y 2
(c) t2 +7t+12
t 2 +5t+4
x 2 +2x+1
(e)
x 2 −2x+1
4 Simplify the following:
(a)
(b)
x +1
x +3 × x +3
x +2
4
x 2 −1 × x +1
6
(b)
(d)
5x 2 +5
10x −10
x 2 −1
x 3 −2x 2 +x
(c)
(d)
(e)
x 2 +3x
x 3 +2x 2 × x2 +4x+4
4x
4xt +4t
xt 2 −t 2 × 4x2 −4
8x+8
x 2 +2x−15
x 2 +4x−5 × x2 +3x−4
x 2 −4x+3
5 Express asasingle fraction
(a)
(b)
(c)
(d)
(e)
3
x +6 + 2
x +1
4
x +2 − 2
(x +2) 2
2x+1
x 2 +x+1 + 4
x −1
x 2 +3x−18
x 2 +7x+6 − 2x2 +7x−4
x 2 +9x+20
3(x +1) 2(x −1)
x 2 +
+4x+4 x 2 −4
Solutions
1 (a) proper (b) proper (c) improper
(d) improper (e) improper (f) improper
2 (a) proper (b) improper (c) improper
(d) improper (e) proper
(f) proper
(g) improper (h) improper (i) proper
(j) proper
y 2 +2
3 (a)
2 −y
x +1
(d)
x(x −1)
(b)
(e)
x 2 +1
(c) t +3
2x−2 t +1
x 2 +2x+1
x 2 −2x+1
4 (a)
(c)
5 (a)
(c)
(d)
x +1
x +2
(x +2)(x +3)
4x 2
5x+15
(x +1)(x +6)
(b)
(d)
6x 2 +3x+3
(x−1)(x 2 +x+1)
−x 2 +x−14
(x +1)(x +5)
(b)
(e)
2
3(x −1)
2(x +1)
t
4x+6
(x +2) 2
(e)
5x 2 −x−10
(x+2) 2 (x−2)
x +4
x −1
1.6 SOLUTIONOFINEQUALITIES
Aninequality isany expression involving one ofthe symbols >, , <, .
a >bmeansaisgreaterthanb
a <bmeansaislessthanb
a bmeansaisgreaterthanorequal tob
a bmeansaislessthanorequal tob
Just as with an equation, when we add or subtract the same quantity to both sides of an
inequalitythe inequalitystill remains.Mathematically wehave