[mathjax]
Exercise 12.1
Question 1
If x is a non-negative integer then solve
2-3x<5-4x
Sol :
Adding both sides 4x
2-3x+4x<5-4x+4x
2+x<5
Subtracting both sides 2
2+x-2<5-2
x<3
x={0,1,2}
[]
ALTERNATE METHOD
2-3x<5-4x
-3x+4x<5-2
x<2
Question 2
Solve 2(x-2)<3x-2, $x \in\{-2,-1,0,1,2,3,4\}$
Also show the solution set on the number line
Sol :
2(x-2)<3x-2
2x-4<3x-2
Adding both sides 4
2x<3x+2
Subtracting both sides 3x
2x-3x<3x+2-3x
-x<2
Multiplying both sides -1
x>-2
x={-1,0,1,2,3,4}
[]
Question 3
Solve (a) 30x<200 (b) 5x-3<3x+1 when
(i) x is integer
(ii) x is a natural number
Sol :
30x<200
Dividing both sides by 30
$\dfrac{30 x}{30}<\dfrac{200}{30}$
$x<\dfrac{20}{3}$
$x<6 \dfrac{2}{3}$
(i)
x={...-3,-2,-1,0,1,2,3,4,5,6.}
(ii)
x={1,2,3,4,5,6}
Question 4
Solve
(a) 3x+5<x-7 (b) 5x-3<3x+1 when
(i) x is integer
(ii) x is a real number
Sol :
3x+5<x-7
Subtracting both sides by 5
3x+5-5<x-7-5
3x<x-12
Subtracting both sides x
2x<-12
Dividing both sides by x
$\dfrac{2x}{x}<\dfrac{-12}{2}$
x<-6
(i)
x={...-9,-8,-7}
(ii)
xϵ(-∞,-6)
Question 7
Solve the following inequations :
(i) 2-3x≥2(x+6)
Sol :
2-3x≥2x+12
Subtracting both side 2x
2-5x≥12
Subtracting both side 2
-5x≥10
Dividing both sides by -5
$\frac{-5 x}{-5} \leq \frac{10}{-5}$
xϵ(-∞,-2)
(ii) 2(2x+3)-10<6(x-2)
Sol :
4x+6-10<6x-12
4x-4<6x-12
Adding both side 4
4x-4+4<6x-12+4
4x<6x-8
Subtracting 6x both sides
-2x<-8
Dividing with -2 both sides
$\frac{-2x}{-2}<\frac{-8}{-2}$
x>4
(iii) -(x-3)+4>-2x+5
Sol :
-x+3+4>-2x+5
-x+7>-2x+5
-x+7>-2x+5
Subtracting both sides with 7
-x>-2x-2
Adding both sides 2x
-x+2x>-2x-2+2x
x>-2
xϵ(-2,∞)
Question 8
Solve the following inequations:
(i) $\frac{3(x-2)}{5} \geq \frac{5(2-x)}{3$
Sol :
$15 \times \frac{3(x-2)}{5} \geq 15 \times \frac{5(2-x)}{3}$
9(x-2)≥25(2-x)
9x-18≥50-25x
Adding both sides 25x
9x-18+25x≥50-25x+25x
34x-18≥50
Adding both sides 18
34x≥68
Dividing both sides by 34
$\frac{34 x}{34}\ge\frac{68}{34}$
x≥2
xϵ(2,∞)
(ii) $\frac{11-2 x}{5} \geq \frac{9-3 x}{8}+\frac{3}{4}, x \in \mathrm{N}$
Sol :
$\frac{11-2 x}{5} \geq \frac{9-3 x+6}{8}$
$\frac{11-2 x}{5} \geq \frac{15-3 x}{8}$
Multiplying both sides by 40
$\frac{40(11-22)}{5} \geq 40\left(\frac{15-3 x}{8}\right)$
88-16x≥75-15x
Adding both sides 15x
88-16x+15x≥75-15x+15x
Subtracting 88 both sides
-x≥-13
Multiplying both sides by -1
x={1,2,3,4,.....13}
Question 9
Solve the following inequations
(i) $\frac{3}{x-2}<0$
Sol :
[]
∴ x-2<0
Adding both sides 2
x-2+2<0+2
x<2
[]=(-∞,2)
(ii) $-\frac{1}{x+2}>0$
Sol :
[]
∴ x+2<0
Subtracting both sides 2
Question 10
Solve the following inequations :
(i) $\frac{x-3}{x+5}>0$
Sol :
=x-3>0 and x+5>0
or
x-3<0 and x+5<0
=x>3 and x>-5
or
x<3 and x<-5
=x>3
or
x<-5
(ii) $\frac{x-3}{x-5}>0$
Sol :
x-3>0 and x-5>0
or
x-3<0 and x-5<0
=x>3 and x>5
or
x<3 and x<5
=x>5
or
x<3
[]
Question 11
Solve the following inequations
(i) $\frac{x-1}{x-3}<1$
Sol :
Subtracting both sides with 1
$\frac{x-1}{x-3}-1<1-1$
$\frac{x-1-(x-3)}{x-3}<0$
$\frac{x-1-x +7}{x-3}<0$
$\frac{2}{x-3}<0$
[]
∴ x-3<0
x<3
[]=(-∞,3)
(ii) $\frac{x+1}{x-7} \geq 2$
Sol :
Subtracting both sides 2
$\frac{x+1}{x-7}-2 \geq 2-2$
$\frac{x+1-2(x-7)}{x-7} \geq 0$
$\frac{x+1-2 x+14}{x-7} \geq 0$
$\frac{-x+15}{x-7} \geq 0$
=-x+15≥0 and x-7>0
or
-x+15≤0 and x-7<0
=x≤15 and x>7
or
x≥15 and x<7
=7<x≤15
or
not possible
[]= (7,15]