Linear Inequations Class 10 RS Aggarwal Goyal Brothers Prakashan ICSE foundation Maths Solutions Exe-4. Step by step solutions of exercise-4 questions as latest prescribe guideline for upcoming exam. Visit official Website CISCE for detail information about ICSE Board Class-10.
Linear Inequations Class 10 RS Aggarwal ICSE foundation Maths Solutions Exe-4
Goyal Brothers Prakashan
| Board | ICSE |
| Subject | Maths |
| Class | 10th |
| Chapter-4 | Linear Inequations |
| Writer | RS Aggarwal |
| Topics | Solution of Exe-4 Questions |
| Edition | 2024-2025 |
How to Solve Linear Inequality Problems and Represent Solutions Set on Number Line
Linear Inequations Class 10 RS Aggarwal Goyal Brothers Prakashan ICSE foundation Maths Solutions Exe-4
Exercise- 4
Page- 41
Solve each of the inequations given below and represent its solution set on a number line :
Que-1: 2x-7 < 4, x ∈ {1,2,3,4,5,6,7}
Solution- 2x-7 < 4,
2x < 4+7
2x < 11
x < 11/2
x < 5.5
So x = (1,2,3,4,5) from solution set given
Now on number line
Que-2: 2x-3 > 3, x ∈ (1,2,3,4,5,6)
Solution- 2x-3 > 3
2x > 3+3
2x > 6
x > 3
So, x = (4,5,6) from solution set given
Now on the number line
Que-3: 9 ≤ 1-2x, x ∈ {-3,-4,-5,-6}
Solution- 9 ≤ 1-2x
2x ≤ 1-9
2x ≤ -8
x ≤ -4
So, x = (-4,-5,-6) from the given solution
Now on the number line
Que-4: (3x-5)/6 > 1/2, x ∈ {0,1,2,3,4,5,6}
Solution- (3x-5)/6 > 1/2
3x-5 > 6/2
3x-5 > 3
3x > 3+5
3x > 8
x > 8/3
x > 2.66..
So, x = (3,4,5,6) from the given solution
Now on the number line
Que-5: 7x-4(3-x) ≥ 3(2x-5), x ∈ {-3,-2,-1,0,1,2,3}
Solution- 7x-4(3-x) ≥ 3(2x-5)
7x-12+4x ≥ 6x-15
11x-12 ≥ 6x-15
11x-6x ≥ -15+12
5x ≥ -3
x ≥ -3/5
x ≥ -0.6
So, x = (0,1,2,3) from the given solution
Now on the number line
Que-6: 4-3x ≥ 3x-14, x ∈ N
Solution- 4-3x ≥ 3x-14
-3x-3x ≥ -14-4
-6x ≥ -18
6x ≤ 18
x ≤ 3
So, x = (1,2,3) from the given solution
Now on the number line
Que-7: 6-5x > 3-4x, x ∈ W
Solution- 6-5x > 3-4x
-5x+4x > 3-6
-x > -3
x < 3
So, x = (0,1,2) from the given solution
Now on the number line
Que-8: (3x/5) – (2x-1)/3 > 1, x ∈ I
Solution- (3x/5) – (2x-1)/3 > 1
LCM of 5 and 3, multiply by 15
15 X (3x/5) – 15 X [(2x-1)/3] > 1 X 15
9x – 5(2x-1) > 15
9x-10x+5 > 15
-x > 15-5
-x > 10
x < -10
So, x = (-11,-12,-13….) from the given solution
Now on the number line
Que-9: 2x + 7/2 > (5x/3) + 3, x ∈ I.
Solution- 2x + 7/2 > (5x/3) + 3
LCM of 2 and 3, multiplying by 6
6 x 2x + 6 x 7/2 > 6 x (5x/3) + 6 x 3
12x + 21 > 10x + 18
12x-10x > 18-21
2x > -3
x > -3/2
x > -1.5
So, x = (0,1,2,3,4….) from the given solution
Now on the number line
Que-10: 2x+3 ≤ 3x+1, x ∈ R
Solution- 2x+3 ≤ 3x+1
2x-3x ≤ 1-3
-x ≤ -2
x ≥ 2
So, x = (x>2, x ∈ R) from the given solution
Now on the number line
Que-11: (5x-8)/3 ≥ (4x-7)/2, x ∈ R
Solution- (5x-8)/3 ≥ (4x-7)/2
(5x-8) x 2 ≥ (4x-7) x 3
10x-16 ≥ 12x-21
10x-12x ≥ -21+16
-2x ≥ -5
2x ≤ 5
x ≤ 5/2
x ≤ 2.5
So, x = {x : x ≤ 5/2, x ∈ R}
Now on the number line
Solve each of the inequations given below. Write the solution set and represent it on the number line :
Que-12: -3 < 2x-1 < x+4, x ∈ I
Solution- -3 < 2x-1 < x+4
-3 < 2x-1 and 2x-1 < x+4
-3+1 < 2x and 2x-x < 4+1
-2 < 2x and x < 5
x > -2/2
x > -1
-1 < x < 5
So, the solution set is {0,1,2,3,4}
Now, on the number line
Que-13: 2+4x < 2x-5 < 3x, x ∈ I
Solution- 2+4x < 2x-5 < 3x
2+4x < 2x-5 and 2x-5 < 3x
4x-2x < -5-2 and -5 < 3x-2x
2x < -7 and -5 < x
x < -3.5 and -5 < x
-5 < x < -3.5
Hence the solution set is {-4}
Now, on the number line.
Que-14: 2 ≤ 2x-3 ≤ 5, x ∈ R
Solution- 2 ≤ 2x-3 ≤ 5
2 ≤ 2x-3 and 2x-3 ≤ 5
2+3 ≤ 2x and 2x ≤ 5+3
5 ≤ 2x and 2x ≤ 8
2.5 ≤ x and x ≤ 4
2.5 ≤ x ≤ 4
Hence the solution set is 2.5 ≤ x ≤ 4
Now, on the number line
Que15: -1 ≤ 3+4x < 23, x ∈ R
Solution- -1 ≤ 3+4x < 23
-1 ≤ 3+4x and 3+4x < 23
-1-3 ≤ 4x and 4x < 23-3
-4 ≤ 4x and 4x < 20
-1 ≤ x and x < 5
-1 ≤ x < 5
Hence the solution set is: {-1 ≤ x < 5 x ∈ R}
Now on the number line
Que-16: -2 ≤ (1/2) – (2x/3) < 1*(5/6), x ∈ R
Solution- -2 ≤ (1/2) – (2x/3) < 1*(5/6)
-2 ≤ 1/2 – (2x/3) and 1/2 – (2x/3) < 11/6
2x/3 ≤ 1/2 + 2 and 1/2 – 11/6 < 2x/3
2x/3 ≤ 5/2 and -8/6 < 2x/3
x ≤ 15/4 and -2 < x
x ≤ 3.75 and -2 < x
-2 < x ≤ 3.75
Hence the solution set is : {-1,0,1,2,3}
Now on the number line :
Que-17: -2/3 < 1+(x/3) ≤ 2/3, x ∈ R
Solution- -2/3 < 1+(x/3) ≤ 2/3
-2/3 < 1+(x/3) and 1+(x/3) ≤ 2/3
(-2/3) – 1 < x/3 and x/3 ≤ (2/3) – 1
-5/3 < x/3 and x/3 ≤ -1/3
-5 < x and x ≤ -1
-5 < x ≤ -1
Hence the solution set is {x : -5 < x ≤ -1}
Now on the number line
Que-18: 2x-5 ≤ 5x+4 < 11, x ∈ R
Solution- 2x-5 ≤ 5x+4 < 11
2x-5 ≤ 5x+4 and 5x+4 < 11
-5-4 ≤ 5x-2x and 5x < 11-4
-9 ≤ 3x and 5x < 7
-3 ≤ x and x < 7/5
-3 ≤ x and x < 7/5
-3 ≤ x < 7/5
Hence the solution set is {x : -3 ≤ x < 7/5}
Now on the number line
Que-19: 1 ≥ 15-7x > 2x-27, x ∈ N
Solution- 1 ≥ 15-7x > 2x-27
1 ≥ 15-7x and 15-7x > 2x-27
7x ≥ 15-1 and 15+27 > 2x+7x
7x ≥ 14 and 42 > 9x
x ≥ 14/7 and 42/9 > x
x ≥ 2 and 14/3 > x
2 ≤ x < 4.66
Hence the solution set is {2,3,4}
Now on the number line
Que-20: -8*(1/2) < (-1/2) – 4x ≤ 7*(1/2), x ∈ I
Solution- -8*(1/2) < (-1/2) – 4x ≤ 7*(1/2)
-17/2 < (-1/2) – 4x and (-1/2) – 4x ≤ 15/2
4x < (-1/2)+(17/2) and (-1/2) – (15/2) ≤ 4x
4x < 16/2 and -16/2 ≤ 4x
4x < 8 and -8 ≤ 4x
x < 2 and -2 ≤ x
-2 ≤ x < 2
Hence the solution set is {-2,-1,0,1}
Now on the number line
Que-21: -2*(2/3) ≤ x+(1/3) < 3*(1/3), x ∈ R
Solution- -2*(2/3) ≤ x+(1/3) < 3*(1/3)
(-8/3) ≤ x+(1/3) and x+(1/3) < 10/3
-(8/3)-(1/3) ≤ x and x < (10/3)-1/3
-9/3 ≤ x and x < 9/3
-3 ≤ x and x < 3
-3 ≤ x < 3
Hence the solution set is {x : -3 ≤ x < 3 x ∈ R}
Now on the number line
Que-22: 5x-11 ≤ 7x-5 < 9, x ∈ R
Solution- 5x-11 ≤ 7x-5 < 9
5x-11 ≤ 7x-5 and 7x-5 < 9
-11+5 ≤ 7x-5x and 7x < 9+5
-6 ≤ 2x and 7x < 14
-3 ≤ x and x < 2
-3 ≤ x < 2
Hence the solution set is {x : -3 ≤ x < 2 x ∈ R}
Now on the number line
Que-23: 2x-1 ≥ x+(7-x)/3 > 2, x ∈ R
Solution- 2x-1 ≥ x+(7-x)/3 > 2
2x-1 ≥ x+(7-x)/3 and x+(7-x)/3 > 2
2x-x ≥ 1+[(7-x)/3] and (3x+7-x)/3 > 2
x ≥ (3+7-x)/3 and 2x+7 > 6
3x ≥ 10-x and 2x > 6-7
3x+x ≥ 10 and 2x > -1
4x ≥ 10 and x > -1/2
x ≥ 5/2 and x > -1/2
Hence the solution set is {x ≥ 5/2 and x > -1/2}
Now on the number line
Que-24: -3+x ≤ (8x/3)+2 ≤ 14/3+2x, x ∈ I
Solution- -3+x ≤ (8x/3)+2 ≤ 14/3+2x
-3+x ≤ (8x/3)+2 and (8x/3)+2 ≤ 14/3+2x
-3+x ≤ (8x+6)/3 and (8x+6)/3 ≤ (14+6x)/3
-9+3x ≤ 8x+6 and 8x+6 ≤ 14+6x
-9-6 ≤ 8x-3x and 8x-6x ≤ 14-6
-15 ≤ 5x and 2x ≤ 8
-3 ≤ x and x ≤ 4
-3 ≤ x ≤ 4
Hence the solution set is {-3,-2,-1,0,1,2,3,4}
Now on the number line
Que-25: -2*(5/6) < (1/2)-(2x/3) ≤ 2, x ∈ W
Solution- -2*(5/6) < (1/2)-(2x/3) ≤ 2
-2*(5/6) < (1/2)-(2x/3) and (1/2)-(2x/3) ≤ 2
-17/6 < (1/2)-(2x/3) and (1/2)-(2x/3) ≤ 2
(2x/3) < (1/2)+(17/6) and (1/2)-2 ≤ (2x/3)
(2x/3) < (3+17)/6 and (1-4)/2 ≤ (2x/3)
(2x/3) < 20/6 and -3/2 ≤ (2x/3)
4x < 20 and -1 ≤ x
x < 5 and -1 ≤ x
-1 ≤ x < 5
Hence the solution set is {-1,0,1,2,3,4}
Now on the number line
Que-26: -5(x-9) ≥ 17-9x > x+2, x ∈ R
Solution- -5(x-9) ≥ 17-9x > x+2
-5(x-9) ≥ 17-9x and 17-9x > x+2
-5x+45 ≥ 17-9x and 17-2 > x+9x
-5x+9x ≥ 17-45 and 15 > 10x
4x ≥ -28 and 15 > 10x
x ≥ -7 and 1.5 > x
-7 ≤ x < 1.5.
Que-27: (-x/3) ≤ (x/2)-1*(1/3) < 1/6, x ∈ R
Solution- (-x/3) ≤ (x/2)-1*(1/3) < 1/6
(-x/3) ≤ (x/2)-1*(1/3) and (x/2)-1*(1/3) < 1/6
(-x/3) ≤ (x/2)-(4/3) and (x/2)-(4/3) < 1/6
4/3 ≤ (x/2)+(x/3) and (x/2) < (1/6)+(4/3)
4/3 ≤ (3x+2x)/6 and x/2 < (1+8)/6
4/3 ≤ 5x/6 and x/2 < 9/6
4/3 x 6/5 ≤ x and x < (9/6) x 2
8/5 ≤ x and x < 3
8/5 ≤ x < 3
Hence the solution set is {1.6 ≤ x < 3}
Now on the number line
Que-28: 4x-19 < (3x/5)-2 ≤ (-2/5)+x, x ∈ R
Solution- 4x-19 < (3x/5)-2 ≤ (-2/5)+x
4x-19 < (3x/5)-2 and (3x/5)-2 ≤ (-2/5)+x
4x-19 < (3x-10)/5 and (3x-10)/5 ≤ (-2+5x)/5
20x-95 < 3x-10 and 3x-10 ≤ -2+5x
20x-3x < -10+95 and -10+2 ≤ 5x-3x
17x < 85 and -8 ≤ 2x
x < 5 and -4 ≤ x
-4 ≤ x < 5
Hence the solution set is {-4 ≤ x < 5}
Now on the number line
Que-29: 2y-3 < y+1 ≤ 4y+7, y ∈ R
Solution- 2y-3 < y+1 ≤ 4y+7
2y-3 < y+1 and y+1 ≤ 4y+7
2y-y < 1+3 and 1-7 ≤ 4y-y
y < 4 and -6 ≤ 3y
y < 4 and -2 ≤ y
-2 ≤ y < 4
Hence the solution set is {-2 ≤ y < 4}
Now on the number line
Que-30: -2+10x ≤ 13x+10 < 24+10x, x ∈ Z
Solution- -2+10x ≤ 13x+10 < 24+10x
-2+10x ≤ 13x+10 and 13x+10 < 24+10x
-2-10 ≤ 13x-10x and 13x-10x < 24-10
-12 ≤ 3x and 3x < 14
-4 ≤ x and x < 14/3
-4 ≤ x < 4.66
Hence the solution set is {-4 ≤ x < 4.66}.
Que-31: Solve the following inequation and write down the solution set : 11x-4 < 15x+4 ≤ 13x+14, x ∈ W Represent the solution set on a real number line.
Solution- 11x – 4 <15x + 4 or 15x + 4 ≤ 13x +14
4x >-8 or 2x ≤ 10
x>-2 or x ≤ 5
∴ x ∈ [-1 ,0,1,2,3,4,5]
Que-32: Given : P = {x : 5 < 2x-1 ≤ 11, x ∈ R} and Q = {x : -1 ≤ 3+4x < 23, x ∈ I}. Represent P and Q on the number line. Find P ∩ Q.
Solution- P = {x : 5 < 2x – 1 ≤ 11}
5 < 2x – 1 ≤ 11
5 < 2 x – 1 and 2x – 1 ≤ 11
– 2 x < – 5 – 1 and 2 x ≤ 11 + 1
– 2x < – 6 and 2x ≤ 12
–x < –3
x > 3 or 3 < x
∴ Solution set = 3 < x ≤ 6 – {4, 5, 6}
Solution set on number line.
Q = {–1 ≤ 3 + 4x < 23}
–1 ≤ 3 + 4 x < 23
–1 < 3 + 4x and 3 + 4 x < 23
–4x < 3 + 1 and 4x < 23 – 3
–4x < 4 and 4x < 20
–x < 1 and x < 5
x > – 1
–1 < x
∴ Solution set = {0, 1, 2, 3, 4}
∴ Solution set on number line
P ∩ Q = {4}.
Que-33: Let A = {x ∈ R : 11x-5 > 7x+3} and B = {x ∈ R : 8x-9 ≥ 15+2x}.
Find A ∩ B and represent it on the number line.
Solution- A = {x: 11x – 5 > 7x + 3, x ∈ R}
= {x: 4x > 8, x ∈ R}
= {x: x > 2, x ∈ R}
B = {x: 18x – 9 ≥ 15 + 12x, x ∈ R}
= {x: 6x ≥ 24, x ∈ R}
= {x: x ≥ 4, x ∈ R}
A ∩ B = {x: x ≥ 4, x ∈ R}
It can be represented on number line as:
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