Exercise 4
Question 1
Solve the inequation 3x - 11 ≤ 3, where x ∈ {1, 2, 3, .........., 10}. Also represent its solution on number line.
Answer
Given,
3x−11≤3⇒3x−11+11≤3+11⇒3x≤14⇒x≤314⇒x≤4.66
But x ∈ {1, 2, 3, ..........,, 10}
∴ Solution Set = {1, 2, 3, 4}.
The graph of the solution set is shown by thick dots on the number line.
Question 2
Solve 2(x - 3) < 1, x ∈ {1, 2, 3, .........., 10}.
Answer
Given,
2(x−3)<1⇒2x−6<1⇒2x−6+6<1+6⇒2x<7⇒x<27⇒x<3.5
But x ∈ {1, 2, 3, ……., 10}
Solution set is {1, 2, 3}.
Question 3
Solve 5 - 4x > 2 - 3x, x ∈ W . Also represent its solution on number line.
Answer
Given,
5−4x>2−3x⇒5−2>−3x+4x⇒3>x⇒x<3
Hence, Solution set is {0, 1, 2}.
The graph of the solution set is shown by thick dots on the number line.
Question 4
List the solution set of 30 – 4(2x – 1) < 30, given that x is a positive integer.
Answer
Given,
30−4(2x−1)<30⇒30−8x+4<30⇒34−8x<30⇒34−30<8x⇒8x>4⇒x>21
Since , x is a positive integer
x = {1, 2, 3, 4, …..}.
Question 5
Solve : 2(x – 2) < 3x – 2, x ∈ { –3, –2, –1, 0, 1, 2, 3} .
Answer
Given,
2(x–2)<3x–2⇒2x–4<3x–2⇒2x–3x<–2+4⇒–x<2⇒x>–2
Solution set = {–1, 0, 1, 2, 3} .
Question 6
If x is a negative integer, find the solution set of 32+31(x+1)>0.
Answer
Given,
32+31(x+1)>0⇒32+3x+31>0⇒3x+1>0⇒3x>–1⇒x>−3
x is a negative integer Solution set = {-2, –1}.
Question 7
Solve x – 3(2 + x) > 2(3x - 1), x ∈ {-3, -2, -1, 0, 1, 2}. Also represent its solution on the number line.
Answer
Given,
x−3(2+x)>2(3x−1)⇒x−6−3x>6x−2⇒x−3x−6x>−2+6⇒−8x>4⇒8x<−4⇒x<−21
Since, x ∈ { -3, -2, -1, 0, 1, 2}
Hence, Solution set = { -3, -2, -1}.
The graph of the solution set is shown by thick dots on the number line.
Question 8
Given x ∈ {1, 2, 3, 4, 5, 6, 7, 9} solve x – 3 < 2x – 1.
Answer
Given,
x−3<2x−1⇒x−2x<−1+3⇒−x<2⇒x>−2
Since, x ∈ {1, 2, 3, 4, 5, 6, 7, 9}
Solution set = {1, 2, 3, 4, 5, 6, 7, 9}.
Question 9
List the solution set of the inequation
21+8x>5x−23,x∈Z
Answer
Given, 21+8x>5x−23⇒8x−5x>−23−21⇒3x>−24⇒3x>−2⇒x>−32
Since , x ∈ Z
Solution set = {0, 1, 2, 3, 4…..}
Question 10
List the solution set of:
511−2x≥89−3x+43,x∈N.
Answer
Given,
511−2x≥89−3x+43⇒511−2x≥89−3x+6⇒8(11−2x)≥5(15−3x)⇒88−16x≥75−15x⇒−16x+15x≥75−88⇒−x≥−13⇒x≤13
Since, x ∈ N
Solution set = {1, 2, 3, 4, 5, ..... , 13}.
Question 11
Find the values of x, which satisfy the inequation:
−2≤21−32x≤165, x ∈ N.
Graph the solution set on the number line.
Answer
Given,
−2≤21−32x≤165⇒−2≤21−32x≤611⇒−2−21≤21−32x−21≤611−21By subtracting 21 on both sides of inequality in the above line.⇒−25≤−32x≤68⇒−25×6≤−32x×6≤68×6 (Multiplying complete equation with 6) ⇒−15≤−4x≤8⇒15≥4x≥−8⇒415≥x≥−2
Since x ∈ N,
∴ Solution Set = {1, 2, 3}
The graph of the solution set is shown by thick dots on the number line.
Question 12
If x ∈ W, find the solution set of 53x−32x−1> 1. Also graph the solution set on the number line, if possible.
Answer
Given,
53x−32x−1>1⇒159x−10x+5>1 [Taking LCM as 15]−x+5>15⇒–x>15–5⇒–x>10⇒x<–10
But x ∈ W
Solution set = Φ
Hence it can't be represented on number line.
Question 13(i)
Solve:
2x+5≤3x+6 where x is a positive odd integer.
Answer
Given,
2x+5≤3x+6⇒2x−3x≤6−5⇒63x−2x≤1⇒6x≤1⇒x≤6
Since, x is a positive odd integer
Hence, x = {1, 3, 5}.
Question 13(ii)
Solve:
32x+3≥43x−1 where x is positive even integer.
Answer
Given,
32x+3≥43x−1⇒32x+33≥43x−41⇒32x+1≥43x−41⇒32x−43x≥−41−1⇒128x−9x≥−45⇒−12x≥−45⇒12x≤45⇒x≤45×12⇒x≤15
Since, x is a positive even integer.
x = {2, 4, 6, 8, 10, 12, 14}.
Question 14
Given that x ∈ I, solve the inequation and graph the solution on the number line :
3≥2x−4+3x≥2
Answer
Given,
3≥2x−4+3x≥2 Solving left side: 3≥2x−4+3x⇒3≥+63x−12+2x⇒3≥+65x−12⇒18≥5x−12⇒5x−12≤18⇒5x≤30⇒x≤6Solving right side:2x−4+3x≥2⇒63x−12+2x≥2⇒65x−12≥2⇒5x−12≥12⇒5x≥24⇒x≥524⇒x≥454
∴ Solution Set = {5, 6}.
The graph of the solution set is shown by thick dots on the number line.
Question 15
Solve 1 ≥ 15 - 7x > 2x - 27, x ∈ N.
Answer
Given,
1≥15−7x>2x−27⇒1≥15−7x and 15−7x>2x−27⇒7x≥15−1 and −7x−2x>−27−15⇒7x≥14 and −9x>−42⇒x≥2 and −x>−942⇒x≥2 and x<314⇒2≤x<314
Since x ∈ N,
Solution set = {2, 3, 4}.
Question 16
If x ∈ Z, solve 2 + 4x < 2x – 5 ≤ 3x. Also represent its solution on the number line.
Answer
Given,
2+4x<2x−5≤3x⇒2+4x<2x−5 and 2x−5≤3x⇒4x−2x<−5−2 and 2x−3x≤5⇒2x<−7 and −x≤5⇒x<−27 and x≥−5⇒−5≤x<−27
Since x ∈ Z,
∴ Solution set = {-5, -4}.
The graph of the solution set is shown by thick dots on the number line.
Question 17
Solve : 34x−10≤25x−7,x∈R and represent the solution set on the number line.
Answer
Given,
34x−10≤25x−7⇒34x−10×6≤25x−7×6 ( Multiplying both sides by 6) ⇒8x−20≤15x−21
⇒8x−15x≤−21+20⇒−7x≤−1⇒7x≥1⇒x≥71
∴ Solution Set = {x : x ∈ R, x ≥ 71}
The graph of the solution set is represented by thick black line starting from and including 71 on the number line.
Question 18
Solve 53x−32x−1>1,x∈R and represent the solution set on the number line.
Answer
Given,
53x−32x−1>1⇒9x−(10x−5)>15⇒9x−10x+5>15⇒−x>15−5⇒−x>10⇒x<−10
x ∈ R.
Hence , Solution set = {x : x ∈ R, x < –10}.
The graph of the solution set is represented by thick black line starting from -10 (not including -10) on the number line.
Question 19
Given that x ∈ R, solve the following inequation and graph the solution on the number line:
-1 ≤ 3 + 4x < 23
Answer
Given,
−1≤3+4x<23⇒–1–3≤4x<23–3⇒–4≤4x<20⇒–1≤x<5,x∈R
Solution Set = {x : x ∈ R, -1 ≤ x < 5}
The graph of the solution set is represented by thick black line starting from -1 ( including -1) till 5 ( not including 5 ) on the number line.
Question 20
Solve the following inequation and graph the solution on the number line.
−232≤x+31<3+31, x ∈ R.
Answer
Given, −232≤x+31<3+31⇒−38≤x+31<310
On multiplying the equation by 3 , we get,
⇒−8≤3x+1<10⇒−8−1≤3x+1−1<10−1 (adding -1 to equation)⇒−9≤3x<9⇒−3≤x<3
∴ Solution set = {x : x ∈ R , -3 ≤ x < 3}.
The graph of the solution set is represented by thick black line starting from -3 ( including -3) till 3 ( not including 3 ) on the number line.
Question 21
Solve the following inequation and represent the solution set on the number line :
−3<−21−32x≤65,x∈R.
Answer
First solving the left equation
−3<−21−32x⇒−3<−(21+32x)⇒−(21+32x)>−3⇒−32x>−3+21⇒−32x>2−5⇒32x<25⇒x<415 .......(1)
Now the right side equation
−21−32x≤65⇒−32x≤65+21⇒−32x≤65+3⇒−32x≤68⇒32x≥−68⇒x≥6×2−8×3⇒x≥−2 .......(2)
From 1 and 2 we get,
−2≤x<415
∴ Solution set ={x : x ∈ R , -2 ≤x<415}
The graph of the solution set is represented by thick black line starting from -2 (including -2) till 415 (excluding 415) on the number line.
Question 22
Solving the following inequation, write the solution set and represent it on the number line.
–3(x−7)≥15−7x>3x+1,x∈R.
Answer
Given,
–3(x−7)≥15−7x>3x+1,x∈R
Solving left side
−3(x−7)≥15−7x⇒−3x+21≥15−7x⇒−3x+7x≥15−21⇒4x≥−6⇒x≥−46⇒x≥−23
Solving right side
15−7x>3x+1⇒15−31>3x+7x⇒344>322x⇒22×344×3>x⇒x<2
∴ Solution Set = {x : x ∈ R, −23 ≤ x < 2}
The graph of the solution set is represented by thick black line starting from and including −23 till 2 (not including 2) on the number line.
Question 23
Solve the following inequation, write down the solution set and represent it on the real number line:
-2 + 10x ≤ 13x + 10 < 24 + 10x, x ∈ Z.
Answer
Given,
−2+10x≤13x+10<24+10x Solving left side −2+10x≤13x+10⇒10x−13x≤10+2⇒−3x≤12⇒3x≥−12⇒x≥−4 Solving right side 13x+10<24+10x⇒13x−10x<24−10⇒3x<14⇒x<314
∴ Solution Set = {x : x ∈ Z , -4 ≤ x < 314} = {-4, -3, -2, -1, 0, 1, 2, 3, 4}
The graph of the solution set is represented by thick black dots.
Question 24
Solve the inequation 2x – 5 ≤ 5x + 4 < 11, where x ∈ I. Also represent the solution set on the number line.
Answer
Given,
2x−5≤5x+4<11⇒2x−5≤5x+4 and 5x+4<11⇒2x−5−4≤5x and 5x+4<11⇒2x−9≤5x and 5x<11−4⇒2x−5x≤9 and 5x<7⇒−3x≤9 and x<57⇒3x≥−9 and x<57⇒x≥−3 and x<1.4∴−3≤x<1.4
∴ Solution set = {-3, -2, -1, 0, 1}
The graph of the solution set is represented by thick black dots on the number line.
Question 25
If x ∈ I, A is the solution set of 2(x - 1) < 3x - 1 and B is the solution set of 4x – 3 ≤ 8 + x, find A ∩ B.
Answer
Given,
2(x−1)<3x−1⇒2x−2<3x−1⇒2x−3x<−1+2⇒−x<1⇒x>−1∴A={0,1,2,3,....}
Also,
4x−3≤8+x⇒4x−x≤8+3⇒3x≤11⇒x≤311∴B={.....,−1,0,1,2,3}
∴ A ∩ B = {0, 1, 2, 3}
Question 26
If P is the solution set of -3x + 4 < 2x - 3, x ∈ N and Q is the solution set of 4x - 5 < 12, x ∈ W, find
(i) P ∩ Q
(ii) Q – P
Answer
First finding solution set P
−3x+4<2x−3⇒−3x−2x<−3−4⇒−5x<−7⇒5x>7⇒x>57
∴ P = {2, 3, 4, 5, ..... }
Finding solution set Q
⇒4x−5<12⇒4x<12+5⇒4x<17⇒x<417
∴ Q = {4, 3, 2, 1, 0}
(i) P ∩ Q = {2, 3, 4}
(ii) Q – P = {0, 1}
Question 27
A = {x : 11x - 5 > 7x + 3, x ∈ R } and
B = {x : 18x - 9 ≥ 15 + 12x, x ∈ R }.
Find the range of set A ∩ B and represent it on a number line.
Answer
A=x:11x−5>7x+3,x∈RB=x:18x−9≥15+12x,x∈RNow, A=11x−5>7x+3⇒11x−7x>3+5⇒4x>8⇒x>2,x∈RB=18x−9≥15+12x⇒18x−12x≥15+9⇒6x≥24⇒x≥4∴A∩B=x:x∈R,x≥4
The graph is represented by a thick back line starting from 4 (including 4)
Question 28
Given: P = {x : 5 < 2x - 1 ≤ 11, x ∈ R}
Q = {x : - 1 ≤ 3 + 4x < 23, x ∈ I} where R = {real numbers}, I = {integers}.
Represent P and Q on number line. Write down the elements of P ∩ Q.
Answers
Given,
P = {x : 5 < 2x - 1 ≤ 11, x ∈ R}
⇒5<2x−1≤11
Solving left side,
5<2x−1⇒6<2x⇒3<x⇒x>3
Solving right side,
2x−1≤11⇒2x≤12⇒x≤6
∴ P = {x : x ∈ R, 3 < x ≤ 6}.
The graph of the solution set of P is represented by thick black line starting from 3 (not including 3) till 6 (including 6).
Given,
Q = {x : - 1 ≤ 3 + 4x < 23, x ∈ I}
⇒−1≤3+4x<23⇒−1≤3+4x and 3+4x<23⇒−4x≤3+1 and 4x<20⇒−4x≤4 and x<5⇒−x≤1 and x<5⇒x≥−1 and x<5
∴ Q = {x : x ∈ I, -1 ≤ x < 5} = {-1, 0, 1, 2, 3, 4}.
The graph of the solution set of Q is represented by thick black dots.
∴ P ∩ Q = {4}
Question 29
If x ∈ I, find the smallest value of x which satisfies the inequation
2x+25>35x+2
Answer
Given,
2x+25>35x+2⇒2x−35x>2−25⇒12x−10x>12−15 (Multiplying both sides by 6)⇒2x>−3⇒x>−23
∴ Smallest value of x which satisfies this inequation is -1
Question 30
Given 20 – 5x < 5(x + 8), find the smallest value of x, when
(i) x ∈ I
(ii) x ∈ W
(iii) x ∈ N
Answer
Given,
20−5x<5(x+8)⇒20−5x<5x+40⇒−5x−5x<40−20⇒−10x<20⇒−x<2⇒x>−2
(i) When x ∈ I, then smallest value = -1.
(ii) When x ∈ W, then smallest value = 0.
(iii) When x ∈ N, then smallest value = 1.
Question 31
Solve the following inequation and represent the solution set on the number line:
4x−19<53x−2≤−52+x,x∈R
Answer
Given,
4x−19<53x−2≤−52+x,x∈R4x−19<53x−2 and 53x−2≤−52+x⇒4x−53x<−2+19 and 53x−2≤−52+x⇒520x−3x<17 and −2+52≤x−53x⇒517x<17 and 5−8≤52x⇒5x<1 and x≥−4⇒x<5 and x≥−4⇒−4≤x<5,x∈R
Hence, Solution set = {x : x ∈ R, -4 ≤ x < 5}.
The graph of the solution set is represented by thick line starting from -4 (including -4) till 5 (not including 5).
Question 32
Solve the given inequation and graph the solution on the number line :
2y - 3 < y + 1 ≤ 4y + 7; y ∈ R
Answer
Given,
2y−3<y+1≤4y+7;y∈R. (a) Solving left side 2y−3<y+1⇒2y−y<1+3⇒y<4⇒4>y(b) Solving right side y+1≤4y+7⇒y−4y≤7−1⇒−3y≤6⇒3y≥−6⇒y≥−2
∴ Solution Set = {y : y ∈ R, -2 ≤ y < 4}
The graph of the solution set is represented by a thick black line starting from -2 (including -2) till 4 (not including 4).
Question 33
Solve the inequation and represent the solution set on the number line.
−3+x≤38x+2≤314+2x,where x∈I.
Answer
Solving left side:
−3+x≤38x+2⇒−3−2≤38x−x⇒−3−2≤38x−3x⇒−5≤35x⇒−1≤3x⇒x≥−3
Solving right side:
38x+2≤314+2x⇒38x−2x≤314−2⇒38x−6x≤314−6⇒32x≤38⇒2x≤8⇒x≤4
∴ Solution set = {x : x ∈ I, -3 ≤ x ≤ 4} = {-3, -2, -1, 0, 1, 2, 3, 4}
The graph of the solution set of is represented by thick black dots.
Question 34
Find the greatest integer which is such that if 7 is added to its double, the resulting number becomes greater than three times the integer.
Answer
Let the greatest integer=xAccording to the condition,2x+7>3x⇒2x−3x>−7⇒−x>−7⇒x<7∴Value of x which is greatest = 6
Question 35
One-third of a bamboo pole is buried in mud, one-sixth of it is in water and the part above the water is greater than or equal to 3 metres. Find the length of the shortest pole.
Answer
Let the length of the shortest pole = x metre
Length of pole which is burried in mud = 3x
Length of pole which is in the water = 6x
Given,
x−[3x+6x]≥3⇒x−(62x+x)≥3⇒x−(63x)≥3⇒x−2x≥3⇒2x≥3⇒x≥6
∴ Length of pole which is shortest = 6 meters.
Chapter Test
Question 1
Solve the inequation : 5x - 2 ≤ 3(3 - x) where x ∈ { -2, -1, 0, 1, 2, 3, 4}. Also represent its solution on the number line.
Answer
Given,
5x−2≤3(3−x)⇒5x−2≤9−3x⇒5x+3x≤9+2⇒8x≤11⇒x≤811
Since, x ∈ { -2, -1, 0, 1, 2, 3, 4}.
∴ Solution set ={-2, -1, 0, 1}.
The graph of the solution set is represented by thick black dots.
Question 2
Solve the inequations : 6x - 5 < 3x + 4, x ∈ I
Answer
Given,
6x−5<3x+4⇒6x−3x<4+5⇒3x<9⇒x<3
x ∈ I
∴ Solution Set = {..., -2, -1, 0, 1, 2}.
Question 3
Find the solution set of the inequation x + 5 ≤ 2x + 3 ; x ∈ R. Graph the solution set on the number line.
Answer
Given,
x+5≤2x+3⇒x−2x≤3−5⇒−x≤−2⇒x≥2
∴ Solution set = {x : x ∈ R, x ≥ 2 }.
The graph of the solution set is represented by thick line starting from and including 2.
Question 4
If x ∈ R (real numbers) and -1 < 3 - 2x ≤ 7, find solution set and represent it on a number line.
Answer
Given,
−1<3−2x≤7⇒−1<3−2x and 3−2x≤7⇒2x<3+1 and −2x≤7−3⇒2x<4 and −2x≤4⇒x<2 and −x≤2⇒x<2 or x≥−2
∴ Solution set = {x : x ∈ R, -2 ≤ x < 2}.
The graph of this inequation is represented by thick black line starting from -2 (including -2) till (not including) 2.
Question 5
Solve the inequation :
75x+1−4(7x+52)≤153+73x−1,x∈R.
Answer
Given,
75x+1−4(7x+52)≤153+73x−1,x∈R.
Multiplying both sides by 35
⇒25x+5−4(5x+14)≤56+15x−5⇒25x+5−20x−56≤56+15x−5⇒5x−51≤51+15x⇒5x−15x≤51+51⇒−10x≤102⇒x≥−10102⇒x≥−551
∴ Solution set = {x : x ∈ R, x ≥−551 }.
Question 6
Find the range of values of x, which satisfy 7 ≤ –4x + 2 < 12, x ∈ R. Graph these values of x on the real number line.
Answer
Given,
7 ≤ –4x + 2 < 12
⇒7≤−4x+2 and −4x+2<12
Solving left side,
7≤−4x+2⇒4x≤2−7⇒4x≤−5⇒x≤4−5
Solving right side,
−4x+2<12⇒−4x<12−2⇒−4x<10⇒4x>−10⇒x>2−5
∴ Solution set = {x : x ∈ R, −25 < x ≤ −45}.
The graph of the inequation is represented by thick black line starting from −25 (excluding −25) till −45 (including −45).
Question 7
If x ∈ R, solve 2x−3≥x+31−x>52x. Also represent the solution on the number line.
Answer
Given,
2x−3≥x+31−x>52x
Solving left side,
2x−3≥x+31−x⇒2x−3≥33x+1−x⇒2x−3≥32x+1⇒3(2x−3)≥2x+1⇒6x−9≥2x+1⇒6x−2x≥1+9⇒4x≥10⇒x≥410⇒x≥25
Solving right side,
x+31−x>52x⇒33x+1−x>52x⇒32x+1>52x⇒5(2x+1)>6x⇒10x+5>6x⇒10x−6x>−5⇒4x>−5⇒x>−45
From left side we get x≥25 and from right side we get x>−45
∴ x≥25
∴ Solution set = {x : x ∈ R, x ≥25 }
The graph of the inequation is represented by thick black line starting from 25 (including 25).
Question 8
Find positive integers which are such that if 6 is subtracted from five times the integer then the resulting number cannot be greater than four times the integer.
Answer
Let the positive integer = x
According to the problem,
5x−6<4x⇒5x−4x<6⇒x<6
∴ Solution set = { 1, 2, 3, 4, 5, 6}.
Question 9
Find three smallest consecutive natural numbers such that the difference between one-third of the largest and one-fifth of the smallest is at least 3.
Answer
Let first least natural number = x
then, second number = x + 1
and third number = x + 2
Given,
31(x+2)−51(x)≥3⇒3x−5x≥3−32⇒155x−3x≥39−2⇒152x≥37⇒2x≥37×15⇒2x≥35⇒x≥235⇒x≥1721
Since the three consecutive numbers should be natural numbers
∴ x = 18
x + 1 = 19
x + 2 = 20
Hence, the three smallest consecutive natural numbers are 18, 19, 20