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Chapter 4

Linear Inequations

Class 10 - ML Aggarwal Understanding ICSE Mathematics


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,

3x1133x11+113+113x14x143x4.663x-11 \leq 3 \\[0.5em] \Rightarrow 3x-11 + 11 \leq 3 + 11 \\[0.5em] \Rightarrow 3x \leq 14 \\[0.5em] \Rightarrow x \leq \dfrac{14}{3} \\[0.5em] \Rightarrow x \leq 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.

Solve the inequation  3x - 11  ≤  3, where x ∈  {1, 2, 3, .........., 10} . Also represent its solution on number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 2

Solve 2(x - 3) < 1, x ∈ {1, 2, 3, .........., 10}.

Answer

Given,

2(x3)<12x6<12x6+6<1+62x<7x<72x<3.52(x - 3) \lt 1 \\[0.5em] \Rightarrow 2x-6 \lt 1 \\[0.5em] \Rightarrow 2x-6 + 6 \lt 1 + 6 \\[0.5em] \Rightarrow 2x \lt 7 \\[0.5em] \Rightarrow x \lt \dfrac{7}{2} \\[0.5em] \Rightarrow x \lt 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,

54x>23x52>3x+4x3>xx<35 - 4x \gt 2 - 3x \\[0.5em] \Rightarrow 5 - 2 \gt -3x + 4x \\[0.5em] \Rightarrow 3 \gt x \\[0.5em] \Rightarrow x \lt 3

Hence, Solution set is {0, 1, 2}.

The graph of the solution set is shown by thick dots on the number line.

Solve 5 - 4x > 2 - 3x, x ∈ W . Also represent its solution on number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 4

List the solution set of 30 – 4(2x – 1) < 30, given that x is a positive integer.

Answer

Given,

304(2x1)<30308x+4<30348x<303430<8x8x>4x>1230 - 4(2x - 1) \lt 30 \\[0.5em] \Rightarrow 30-8x+4 \lt 30 \\[0.5em] \Rightarrow 34-8x \lt 30 \\[0.5em] \Rightarrow 34-30 \lt 8x \\[0.5em] \Rightarrow 8x \gt 4 \\[0.5em] \Rightarrow x \gt \dfrac{1}{2}

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(x2)<3x22x4<3x22x3x<2+4x<2x>22(x – 2) \lt 3x – 2 \\[0.5em] \Rightarrow 2x – 4 \lt 3x – 2 \\[0.5em] \Rightarrow 2x – 3x \lt – 2 + 4 \\[0.5em] \Rightarrow – x \lt 2 \\[0.5em] \Rightarrow x \gt – 2 \\[0.5em]

Solution set = {–1, 0, 1, 2, 3} .

Question 6

If x is a negative integer, find the solution set of 23+13(x+1)>0\dfrac{2}{3}+\dfrac{1}{3} (x + 1) \gt 0.

Answer

Given,

23+13(x+1)>023+x3+13>0x3+1>0x3>1x>3\dfrac{2}{3}+\dfrac{1}{3} (x + 1) \gt 0 \\[0.5em] \Rightarrow \dfrac{2}{3} +\dfrac{x}{3} + \dfrac{1}{3} \gt 0 \\[0.5em] \Rightarrow \dfrac{x}{3} + 1 \gt 0 \\[0.5em] \Rightarrow \dfrac{x}{3} \gt – 1 \\[0.5em] \Rightarrow x \gt -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,

x3(2+x)>2(3x1)x63x>6x2x3x6x>2+68x>48x<4x<12x - 3(2 + x) \gt 2(3x - 1) \\[0.5em] \Rightarrow x - 6 - 3x \gt 6x - 2 \\[0.5em] \Rightarrow x - 3x - 6x \gt - 2 + 6 \\[0.5em] \Rightarrow - 8x \gt 4 \\[0.5em] \Rightarrow 8x \lt -4 \\[0.5em] \Rightarrow x \lt -\dfrac{1}{2}

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.

Solve x – 3(2 + x) > 2(3x - 1), x ∈  {-3, -2, -1, 0, 1, 2}. Also represent its solution on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 8

Given x ∈ {1, 2, 3, 4, 5, 6, 7, 9} solve x – 3 < 2x – 1.

Answer

Given,

x3<2x1x2x<1+3x<2x>2x - 3 \lt 2x - 1\\[0.5em] \Rightarrow x - 2x \lt - 1 + 3\\[0.5em] \Rightarrow - x \lt 2\\[0.5em] \Rightarrow x \gt - 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

12+8x>5x32,xZ\dfrac{1}{2} + 8x \gt 5x -\dfrac{3}{2}, x ∈ \bold{Z}

Answer

Given, 12+8x>5x328x5x>32123x>423x>2x>23\dfrac{1}{2} +8x \gt 5x - \dfrac{3}{2} \\[0.5em] \Rightarrow 8x-5x \gt - \dfrac{3}{2}-\dfrac{1}{2} \\[0.5em] \Rightarrow 3x \gt - \dfrac{4}{2} \\[0.5em] \Rightarrow 3x \gt -2 \\[0.5em] \Rightarrow x \gt -\dfrac{2}{3}

Since , x ∈ Z
Solution set = {0, 1, 2, 3, 4…..}

Question 10

List the solution set of:

112x593x8+34,xN.\dfrac{11-2x}{5} \ge \dfrac{9-3x}{8} + \dfrac{3}{4}, x ∈ \bold{N}.

Answer

Given,

112x593x8+34112x593x+688(112x)5(153x)8816x7515x16x+15x7588x13x13\dfrac{11-2x}{5} \ge \dfrac{9-3x}{8} + \dfrac{3}{4} \\[0.5em] \Rightarrow \dfrac{11-2x}{5} \ge \dfrac{9 - 3x + 6}{8} \\[0.5em] \Rightarrow 8(11 - 2x) \ge 5(15 - 3x) \\[0.5em] \Rightarrow 88 - 16x \ge 75 - 15x \\[0.5em] \Rightarrow - 16x + 15x \ge 75 - 88 \\[0.5em] \Rightarrow - x \ge - 13 \\[0.5em] \Rightarrow x ≤ 13 \\[0.5em]

Since, x ∈ N
Solution set = {1, 2, 3, 4, 5, ..... , 13}.

Question 11

Find the values of x, which satisfy the inequation:

2122x3156−2 \le \dfrac{1}{2}-\dfrac{2x}{3} \le 1\dfrac{5}{6}, x ∈ N.

Graph the solution set on the number line.

Answer

Given,

2122x31562122x3116212122x31211612By subtracting 12 on both sides of inequality in the above line.522x38652×62x3×686×6 (Multiplying complete equation with 6) 154x8154x8154x2-2 \le \dfrac{1}{2} - \dfrac{2x}{3} \le 1\dfrac{5}{6} \\[0.5em] \Rightarrow -2 \le \dfrac{1}{2}-\dfrac{2x}{3} \le \dfrac{11}{6} \\[0.5em] \Rightarrow -2-\dfrac{1}{2} \le \dfrac{1}{2} - \dfrac{2x}{3} - \dfrac{1}{2} \le \dfrac{11}{6} - \dfrac{1}{2}\\[0.5em] \text {By subtracting } \dfrac{1}{2} \text{ on both sides of inequality in the above line.}\\[0.5em] \Rightarrow -\dfrac{5}{2} \le -\dfrac{2x}{3} \le \dfrac{8}{6} \\[0.5em] \Rightarrow -\dfrac{5}{2} \times 6 \le -\dfrac{2x}{3}\times 6 \le \dfrac{8}{6}\times 6 \text{ (Multiplying complete equation with 6) } \\[0.5em] \Rightarrow -15 \le -4x \le 8 \\[0.5em] \Rightarrow 15 \ge 4x \ge -8 \\[0.5em] \Rightarrow \dfrac{15}{4} \ge x \ge -2

Since x ∈ N,

∴ Solution Set = {1, 2, 3}

The graph of the solution set is shown by thick dots on the number line.

Find the values of x, which satisfy the inequation -2 ≤ (1/2) - (2x/3) ≤ 1(5/6), x ∈ N. Graph the solution set on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 12

If x ∈ W, find the solution set of 35x2x13>\dfrac{3}{5} x - \dfrac{2x−1}{3} \gt 1. Also graph the solution set on the number line, if possible.

Answer

Given,

35x2x13>19x10x+515>1 [Taking LCM as 15]x+5>15x>155x>10x<10\dfrac{3}{5}x-\dfrac{2x−1}{3} \gt 1\\[0.5em] \Rightarrow \dfrac{9x - 10x + 5}{15} \gt 1 \text{ [Taking LCM as 15]} \\[0.5em] -x + 5 \gt 15 \\[0.5em] \Rightarrow – x \gt 15 – 5\\[0.5em] \Rightarrow – x \gt 10\\[0.5em] \Rightarrow x \lt – 10\\[0.5em]

But x ∈ W
Solution set = Φ
Hence it can't be represented on number line.

Question 13(i)

Solve:

x2+5x3+6\dfrac{x}{2} + 5 \le \dfrac{x}{3} + 6 where x is a positive odd integer.

Answer

Given,

x2+5x3+6x2x3653x2x61x61x6\dfrac{x}{2} +5 \le \dfrac{x}{3} + 6\\[0.5em] \Rightarrow \dfrac{x}{2} - \dfrac{x}{3} \le 6-5\\[0.5em] \Rightarrow \dfrac{3x-2x}{6} \le 1\\[0.5em] \Rightarrow \dfrac{x}{6} \le 1\\[0.5em] \Rightarrow x \le 6

Since, x is a positive odd integer
Hence, x = {1, 3, 5}.

Question 13(ii)

Solve:

2x+333x14\dfrac{2x+3}{3} \ge \dfrac{3x−1}{4} where x is positive even integer.

Answer

Given,

2x+333x142x3+333x4142x3+13x4142x33x41418x9x1254x1254x1254x54×12x15\dfrac{2x+3}{3} \ge \dfrac{3x−1}{4}\\[0.5em] \Rightarrow \dfrac{2x}{3} + \dfrac{3}{3} \ge \dfrac{3x}{4}-\dfrac{1}{4}\\[0.5em] \Rightarrow \dfrac{2x}{3} + 1 \ge \dfrac{3x}{4} - \dfrac{1}{4}\\[0.5em] \Rightarrow \dfrac{2x}{3} - \dfrac{3x}{4} \ge -\dfrac{1}{4}-1\\[0.5em] \Rightarrow \dfrac{8x-9x}{12} \ge - \dfrac{5}{4}\\[0.5em] \Rightarrow -\dfrac{x}{12} \ge -\dfrac{5}{4}\\[0.5em] \Rightarrow \dfrac{x}{12} \le \dfrac{5}{4}\\[0.5em] \Rightarrow x \le \dfrac{5}{4} \times 12\\[0.5em] \Rightarrow x \le 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 :

3x42+x323 \ge \dfrac{x−4}{2} + \dfrac{x}{3} \ge 2

Answer

Given,

3x42+x32 Solving left side: 3x42+x33+3x12+2x63+5x126185x125x12185x30x6Solving right side:x42+x323x12+2x625x12625x12125x24x245x4453 \ge \dfrac{x−4}{2}+ \dfrac{x}{3} \ge 2\\[2em] \text{ Solving left side: }\\[1em] 3 \ge \dfrac{x−4}{2}+ \dfrac{x}{3} \\[0.5em] \Rightarrow 3 \ge + \dfrac{3x-12+2x}{6}\\[0.5em] \Rightarrow 3 \ge + \dfrac{5x-12}{6}\\[0.5em] \Rightarrow 18 \ge 5x-12\\[0.5em] \Rightarrow 5x-12 \le 18\\[0.5em] \Rightarrow 5x \le 30\\[0.5em] \Rightarrow x \le 6\\[1em] \text{Solving right side:} \\[1em] \dfrac{x−4}{2} + \dfrac{x}{3} ≥ 2\\[0.5em] \Rightarrow \dfrac{3x-12+2x}{6} \ge 2 \\[0.5em] \Rightarrow \dfrac{5x-12}{6} \ge 2\\[0.5em] \Rightarrow 5x-12 \ge 12 \\[0.5em] \Rightarrow 5x \ge 24\\[0.5em] \Rightarrow x \ge \dfrac{24}{5}\\[0.5em] \Rightarrow x \ge 4\dfrac{4}{5} \\[0.5em]

∴ Solution Set = {5, 6}.

The graph of the solution set is shown by thick dots on the number line.

Given that x ∈ I, solve the inequation and graph the solution on the number line: 3 ≥ (x - 4)/2 + (x/3) ≥ 2. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 15

Solve 1 ≥ 15 - 7x > 2x - 27, x ∈ N.

Answer

Given,

1157x>2x271157x and 157x>2x277x151 and 7x2x>27157x14 and 9x>42x2 and x>429x2 and x<1432x<1431 \ge 15 - 7x \gt 2x - 27\\[0.5em] \Rightarrow 1 \ge 15 - 7x \text{ and } 15 - 7x \gt 2x - 27\\[0.5em] \Rightarrow 7x \ge 15-1 \text{ and } -7x-2x \gt -27-15\\[0.5em] \Rightarrow 7x \ge 14 \text{ and } -9x \gt -42\\[0.5em] \Rightarrow x \ge 2 \text{ and } -x \gt -\dfrac{42}{9}\\[0.5em] \Rightarrow x \ge 2 \text{ and } x \lt \dfrac{14}{3}\\[0.5em] \Rightarrow 2 \le x \lt \dfrac{14}{3}

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<2x53x2+4x<2x5 and 2x53x4x2x<52 and 2x3x52x<7 and x5x<72 and x55x<722 + 4x \lt 2x - 5 \le 3x\\[0.5em] \Rightarrow 2 + 4x \lt 2x - 5 \text{ and } 2x - 5 \le 3x \\[0.5em] \Rightarrow 4x - 2x \lt - 5 - 2 \text{ and } 2x - 3x \le 5\\[0.5em] \Rightarrow 2x \lt -7 \text{ and } -x \le 5\\[0.5em] \Rightarrow x \lt -\dfrac{7}{2} \text{ and } x \ge -5\\[0.5em] \Rightarrow -5\le x \lt -\dfrac{7}{2}

Since x ∈ Z,

∴ Solution set = {-5, -4}.

The graph of the solution set is shown by thick dots on the number line.

If x ∈ Z, solve 2 + 4x < 2x – 5 ≤ 3x. Also represent its solution on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 17

Solve : 4x1035x72,xR\dfrac{4x-10}{3} \le \dfrac{5x-7}{2}, x ∈ \bold{R} and represent the solution set on the number line.

Answer

Given,

4x1035x724x103×65x72×6 ( Multiplying both sides by 6) 8x2015x21\dfrac{4x−10}{3} \le \dfrac{5x−7}{2}\\[0.5em] \Rightarrow \dfrac{4x-10}{3} \times 6 \le \dfrac{5x-7}{2} \times 6 \text{ ( Multiplying both sides by 6) }\\[0.5em] \Rightarrow 8x - 20 \le 15x - 21

8x15x21+207x17x1x17\Rightarrow 8x-15x \le -21 +20\\[0.5em] \Rightarrow -7x \le -1\\[0.5em] \Rightarrow 7x \ge 1\\[0.5em] \Rightarrow x \ge \dfrac{1}{7}

∴ Solution Set = {x : x ∈ R, x ≥ 17\dfrac{1}{7}}

The graph of the solution set is represented by thick black line starting from and including 17\dfrac{1}{7} on the number line.

Solve : (4x - 10) / 3 ≤ (5x - 7) / 2, x ∈ R and represent the solution set on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 18

Solve 3x52x13>1,xR\dfrac{3x}{5} - \dfrac{2x-1}{3} \gt 1, x ∈ \bold{R} and represent the solution set on the number line.

Answer

Given,

3x52x13>19x(10x5)>159x10x+5>15x>155x>10x<10\dfrac{3x}{5} - \dfrac{2x-1}{3} \gt 1 \\[0.5em] \Rightarrow 9x - (10x - 5) \gt 15 \\[0.5em] \Rightarrow 9x - 10x + 5 \gt 15 \\[0.5em] \Rightarrow - x \gt 15 - 5 \\[0.5em] \Rightarrow - x \gt 10 \\[0.5em] \Rightarrow x \lt -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.

Solve : (3x/5) - (2x - 1) / 3 > 1, x ∈ R and represent the solution set on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 19

Given that x ∈ R, solve the following inequation and graph the solution on the number line:

-1 ≤ 3 + 4x < 23

Answer

Given,

13+4x<23134x<23344x<201x<5,xR-1 \le 3 + 4x \lt 23 \\[0.5em] \Rightarrow – 1 – 3 \le 4x \lt 23 – 3 \\[0.5em] \Rightarrow – 4 \le 4x \lt 20 \\[0.5em] \Rightarrow – 1 \le x \lt 5, x ∈ \bold{R} \\[0.5em]

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.

Given that x ∈ R, solve the following inequation and graph the solution on the number line: -1 ≤ 3 + 4x < 23. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 20

Solve the following inequation and graph the solution on the number line.

223x+13<3+13-2\dfrac{2}{3} \le x + \dfrac{1}{3} \lt 3 + \dfrac{1}{3}, x ∈ R.

Answer

Given, 223x+13<3+1383x+13<103-2 \dfrac{2}{3} \le x + \dfrac{1}{3} \lt 3+ \dfrac{1}{3} \\[0.5em] \Rightarrow -\dfrac{8}{3} \le x + \dfrac{1}{3} \lt \dfrac{10}{3}

On multiplying the equation by 3 , we get,

83x+1<10813x+11<101 (adding -1 to equation)93x<93x<3\Rightarrow -8 \le 3x + 1 \lt 10\\[0.5em] \Rightarrow -8-1 \le 3x + 1 -1 \lt 10 - 1 \text{ (adding -1 to equation)} \\[0.5em] \Rightarrow -9 \le 3x \lt 9 \\[0.5em] \Rightarrow -3 \le x \lt 3 \\[0.5em]

∴ 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.

Solve the following inequation and graph the solution on the number line: -2(2/3) ≤ x + (1/3) < 3 + (1/3), x ∈ R. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 21

Solve the following inequation and represent the solution set on the number line :

3<122x356,xR.-3 \lt - \dfrac{1}{2} -\dfrac{2x}{3} \le \dfrac{5}{6}, x ∈ \bold{R}.

Answer

First solving the left equation

3<122x33<(12+2x3)(12+2x3)>32x3>3+122x3>522x3<52x<154 .......(1)-3 \lt - \dfrac{1}{2} −\dfrac{2x}{3} \\[0.5em] \Rightarrow -3 \lt -(\dfrac{1}{2} + \dfrac{2x}{3}) \\[0.5em] \Rightarrow -\Big(\dfrac{1}{2} + \dfrac{2x}{3}\Big) \gt -3 \\[0.5em] \Rightarrow −\dfrac{2x}{3} \gt -3 + \dfrac{1}{2} \\[0.5em] \Rightarrow −\dfrac{2x}{3} \gt \dfrac{-5}{2} \\[0.5em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{5}{2} \\[0.5em] \Rightarrow x \lt \dfrac{15}{4} \text{ .......(1)}

Now the right side equation

122x3562x356+122x35+362x3862x386x8×36×2x2 .......(2)-\dfrac{1}{2} - \dfrac{2x}{3} \le \dfrac{5}{6}\\[0.5em] \Rightarrow - \dfrac{2x}{3} \le \dfrac{5}{6} + \dfrac{1}{2} \\[0.5em] \Rightarrow - \dfrac{2x}{3} \le \dfrac{5+3}{6}\\[0.5em] \Rightarrow - \dfrac{2x}{3} \le \dfrac{8}{6} \\[0.5em] \Rightarrow \dfrac{2x}{3} \ge -\dfrac{8}{6}\\[0.5em] \Rightarrow x \ge \dfrac{-8 \times 3}{6 \times 2}\\[0.5em] \Rightarrow x \ge -2 \text{ .......(2)}

From 1 and 2 we get,

2x<154-2 \le x \lt \dfrac{15}{4}
∴ Solution set ={x : x ∈ R , -2 x<154\le x \lt \dfrac{15}{4}}

The graph of the solution set is represented by thick black line starting from -2 (including -2) till 154\dfrac{15}{4} (excluding 154\dfrac{15}{4}) on the number line.

Solve the following inequation and represent the solution set on the number line: -3 < -(1/2) - (2x/3) ≤ (5/6) x ∈ R. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 22

Solving the following inequation, write the solution set and represent it on the number line.

3(x7)157x>x+13,xR.–3(x - 7) \ge 15 - 7x \gt \dfrac{x+1}{3}, x ∈ \bold{R}.

Answer

Given,

3(x7)157x>x+13,xR–3(x - 7) \ge 15 - 7x \gt \dfrac{x+1}{3}, x ∈ \bold{R}

Solving left side

3(x7)157x3x+21157x3x+7x15214x6x64x32-3(x-7) \ge 15 - 7x \\[0.5em] \Rightarrow -3x + 21 \ge 15 - 7x \\[0.5em] \Rightarrow -3x + 7x \ge 15 - 21 \\[0.5em] \Rightarrow 4x \ge -6 \\[0.5em] \Rightarrow x \ge -\dfrac{6}{4} \\[0.5em] \Rightarrow x \ge -\dfrac{3}{2}

Solving right side

157x>x+131513>x3+7x443>22x344×322×3>xx<215 -7x \gt \dfrac{x+1}{3} \\[0.5em] \Rightarrow 15- \dfrac{1}{3} \gt \dfrac{x}{3} + 7x \\[0.5em] \Rightarrow \dfrac{44}{3} \gt \dfrac{22x}{3} \\[0.5em] \Rightarrow \dfrac{44 \times 3}{22 \times 3} \gt x \\[0.5em] \Rightarrow x \lt 2

∴ Solution Set = {x : x ∈ R, 32-\dfrac{3}{2} ≤ x < 2}

The graph of the solution set is represented by thick black line starting from and including 32-\dfrac{3}{2} till 2 (not including 2) on the number line.

Solving the following inequation, write the solution set and represent it on the number line: -3(x - 7) ≥ 15 - 7x > (x + 1) / 3, x ∈ R. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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+10x13x+10<24+10x Solving left side 2+10x13x+1010x13x10+23x123x12x4 Solving right side 13x+10<24+10x13x10x<24103x<14x<143−2 + 10x \le 13x + 10 \lt 24 + 10x \\[1em] \text{ Solving left side } \\[0.5em] -2 + 10x \le 13x + 10 \\[0.5em] \Rightarrow 10x -13x \le 10 +2\\[0.5em] \Rightarrow -3x \le 12 \\[0.5em] \Rightarrow 3x \ge -12 \\[0.5em] \Rightarrow x \ge -4 \\[1em] \text { Solving right side } \\[0.5em] 13x + 10 \lt 24 + 10x \\[0.5em] \Rightarrow 13x -10x \lt 24 -10 \\[0.5em] \Rightarrow 3x \lt 14 \\[0.5em] \Rightarrow x \lt \dfrac{14}{3}

∴ Solution Set = {x : x ∈ Z , -4 ≤ x < 143\dfrac{14}{3}} = {-4, -3, -2, -1, 0, 1, 2, 3, 4}

The graph of the solution set is represented by thick black dots.

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. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 24

Solve the inequation 2x – 5 ≤ 5x + 4 < 11, where x ∈ I. Also represent the solution set on the number line.

Answer

Given,

2x55x+4<112x55x+4 and 5x+4<112x545x and 5x+4<112x95x and 5x<1142x5x9 and 5x<73x9 and x<753x9 and x<75x3 and x<1.43x<1.42x - 5 \le 5x + 4 \lt 11 \\[0.5em] \Rightarrow 2x - 5 \le 5x + 4 \text { and } 5x + 4 \lt 11 \\[0.5em] \Rightarrow 2x - 5 - 4 \le 5x \text { and } 5x + 4 \lt 11 \\[0.5em] \Rightarrow 2x - 9 \le 5x \text{ and } 5x \lt 11 - 4 \\[0.5em] \Rightarrow 2x - 5x \le 9 \text{ and } 5x \lt 7 \\[0.5em] \Rightarrow -3x \le 9 \text{ and } x \lt \dfrac{7}{5} \\[0.5em] \Rightarrow 3x \ge - 9 \text{ and } x \lt \dfrac{7}{5} \\[0.5em] \Rightarrow x \ge - 3 \text{ and } x \lt 1.4 \\[0.5em] \therefore -3 \le x \lt 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.

Solve the inequation 2x – 5 ≤ 5x + 4 < 11, where x ∈ I. Also represent the solution set on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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(x1)<3x12x2<3x12x3x<1+2x<1x>1A={0,1,2,3,....}2 (x - 1) \lt 3 x - 1 \\[0.5em] \Rightarrow 2x - 2 \lt 3x - 1 \\[0.5em] \Rightarrow 2x - 3x \lt - 1 + 2 \\[0.5em] \Rightarrow - x \lt 1 \\[0.5em] \Rightarrow x \gt - 1 \\[0.5em] \therefore \text{A} = \lbrace 0, 1, 2, 3, ....\rbrace

Also,

4x38+x4xx8+33x11x113B={.....,1,0,1,2,3}4x - 3 \le 8 + x \\[0.5em] \Rightarrow 4x - x \le 8 + 3 \\[0.5em] \Rightarrow 3x \le 11 \\[0.5em] \Rightarrow x \le \dfrac{11}{3} \\[0.5em] \therefore \text{B} = \lbrace....., -1, 0, 1, 2, 3\rbrace

∴ 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<2x33x2x<345x<75x>7x>75-3 x + 4 \lt 2 x - 3 \\[0.5em] \Rightarrow - 3x - 2x \lt -3-4 \\[0.5em] \Rightarrow - 5x \lt - 7 \\[0.5em] \Rightarrow 5x \gt 7 \\[0.5em] \Rightarrow x \gt \dfrac{7}{5}

∴ P = {2, 3, 4, 5, ..... }

Finding solution set Q

4x5<124x<12+54x<17x<174\Rightarrow 4x-5 \lt 12 \\[0.5em] \Rightarrow 4x \lt 12 + 5 \\[0.5em] \Rightarrow 4x \lt 17 \\[0.5em] \Rightarrow x \lt \dfrac{17}{4}

∴ 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:11x5>7x+3,xRB=x:18x915+12x,xRNow, A=11x5>7x+311x7x>3+54x>8x>2,xRB=18x915+12x18x12x15+96x24x4AB=x:xR,x4\text{A} = {x : 11x - 5 \gt 7x + 3, x ∈ \bold{R}} \\[0.5em] \text{B} = {x : 18x - 9 \ge 15 + 12x, x ∈ \bold{R}} \\[1em] \text{Now, A} = 11x - 5 \gt 7x + 3 \\[0.5em] \Rightarrow 11x - 7x \gt 3 + 5 \\[0.5em] \Rightarrow 4x \gt 8 \\[0.5em] \Rightarrow x \gt 2, x ∈ \bold{R} \\[1em] B= 18x - 9 \ge 15 + 12x \\[0.5em] \Rightarrow 18x -12x \ge 15 + 9 \\[0.5em] \Rightarrow 6x \ge 24 \\[0.5em] \Rightarrow x \ge 4 \\[0.5em] \therefore \text{A} ∩ \text{B} = {x : x ∈ \bold{R}, x \ge 4}

The graph is represented by a thick back line starting from 4 (including 4)

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. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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<2x111\Rightarrow 5 \lt 2x - 1 \le 11

Solving left side,

5<2x16<2x3<xx>35 \lt 2x -1 \\[0.5em] \Rightarrow 6 \lt 2x \\[0.5em] \Rightarrow 3 \lt x \\[0.5em] \Rightarrow x \gt 3 \\[0.5em]

Solving right side,

2x1112x12x62x -1 \le 11 \\[0.5em] \Rightarrow 2x \le 12 \\[0.5em] \Rightarrow x \le 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: 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. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Given,

Q = {x : - 1 ≤ 3 + 4x < 23, x ∈ I}

13+4x<2313+4x and 3+4x<234x3+1 and 4x<204x4 and x<5x1 and x<5x1 and x<5\Rightarrow -1 \le 3 + 4x \lt 23 \\[0.5em] \Rightarrow -1 \le 3 + 4x \text{ and } 3 + 4x \lt 23 \\[0.5em] \Rightarrow -4x \le 3 + 1 \text{ and } 4x \lt 20 \\[0.5em] \Rightarrow -4x \le 4 \text{ and } x \lt 5 \\[0.5em] \Rightarrow -x \le 1 \text{ and } x \lt 5 \\[0.5em] \Rightarrow x \ge -1 \text{ and } x \lt 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.

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. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

∴ P ∩ Q = {4}

Question 29

If x ∈ I, find the smallest value of x which satisfies the inequation

2x+52>5x3+22x +\dfrac{5}{2} \gt \dfrac{5x}{3} + 2

Answer

Given,

2x+52>5x3+22x5x3>25212x10x>1215 (Multiplying both sides by 6)2x>3x>322x + \dfrac{5}{2} \gt \dfrac{5x}{3} + 2 \\[0.5em] \Rightarrow 2x - \dfrac{5x}{3} \gt 2 - \dfrac{5}{2} \\[0.5em] \Rightarrow 12x - 10x \gt 12 - 15 \text{ (Multiplying both sides by 6)} \\[0.5em] \Rightarrow 2x \gt - 3 \\[0.5em] \Rightarrow x \gt - \dfrac{3}{2}

∴ 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,

205x<5(x+8)205x<5x+405x5x<402010x<20x<2x>220 - 5 x \lt 5 (x + 8) \\[0.5em] \Rightarrow 20 - 5x \lt 5x + 40 \\[0.5em] \Rightarrow - 5x - 5x \lt 40 - 20 \\[0.5em] \Rightarrow - 10x \lt 20 \\[0.5em] \Rightarrow - x \lt 2 \\[0.5em] \Rightarrow x \gt - 2 \\[0.5em]

(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:

4x19<3x5225+x,xR4x - 19 \lt \dfrac{3x}{5} - 2 \le - \dfrac{2}{5} + x, x ∈ \bold{R}

Answer

Given,

4x19<3x5225+x,xR4x19<3x52 and 3x5225+x4x3x5<2+19 and 3x5225+x20x3x5<17 and 2+25x3x517x5<17 and 852x5x5<1 and x4x<5 and x44x<5,xR4x-19 \lt \dfrac{3x}{5} - 2 \le - \dfrac{2}{5} + x, x ∈ \bold{R} \\[1em] 4x - 19 \lt \dfrac{3x}{5} -2 \text{ and } \dfrac{3x}{5} - 2 \le - \dfrac{2}{5} + x \\[0.5em] \Rightarrow 4x -\dfrac{3x}{5} \lt -2 + 19 \text{ and } \dfrac{3x}{5} - 2 \le - \dfrac{2}{5} + x \\[0.5em] \Rightarrow \dfrac{20x-3x}{5} \lt 17 \text{ and } -2 + \dfrac{2}{5} \le x - \dfrac{3x}{5} \\[0.5em] \Rightarrow \dfrac{17x}{5} \lt 17 \text{ and } \dfrac{-8}{5} \le \dfrac{2x}{5} \\[0.5em] \Rightarrow \dfrac{x}{5} \lt 1 \text{ and } x \ge -4 \\[0.5em] \Rightarrow x \lt 5 \text{ and } x \ge -4 \\[0.5em] \Rightarrow -4 \le x \lt 5 , x ∈ \bold{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).

Solve the following inequation and represent the solution set on the number line: 4x - 19 < (3x/5) - 2 ≤ -(2/5) + x, x ∈ R. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 32

Solve the given inequation and graph the solution on the number line :

2y - 3 < y + 1 ≤ 4y + 7; y ∈ R

Answer

Given,

2y3<y+14y+7;yR. (a) Solving left side 2y3<y+12yy<1+3y<44>y(b) Solving right side y+14y+7y4y713y63y6y22y - 3 \lt y + 1 \le 4y + 7; y ∈ \bold{R}. \\[0.5em] \text{ (a) Solving left side } \\[0.5em] 2y - 3 \lt y + 1 \\[0.5em] \Rightarrow 2y - y \lt 1 + 3 \\[0.5em] \Rightarrow y \lt 4 \\[0.5em] \Rightarrow 4 \gt y \\[0.5em] \text{(b) Solving right side } \\[0.5em] y + 1 \le 4y + 7 \\[0.5em] \Rightarrow y - 4y \le 7 - 1 \\[0.5em] \Rightarrow -3y \le 6 \\[0.5em] \Rightarrow 3y \ge -6 \\[0.5em] \Rightarrow y \ge -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).

Solve the given inequation and graph the solution on the number line: 2y - 3 < y + 1 ≤ 4y + 7; y ∈ R. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 33

Solve the inequation and represent the solution set on the number line.

3+x8x3+2143+2x,where xI.-3 + x \le \dfrac{8x}{3} + 2 \le \dfrac{14}{3} + 2x, \text{where } x ∈ \bold{I}.

Answer

Solving left side:

3+x8x3+2328x3x328x3x355x31x3x3-3 + x \le \dfrac{8x}{3} + 2 \\[0.5em] \Rightarrow -3 -2 \le \dfrac{8x}{3} -x \\[0.5em] \Rightarrow -3 -2 \le \dfrac{8x-3x}{3} \\[0.5em] \Rightarrow -5 \le \dfrac{5x}{3} \\[0.5em] \Rightarrow -1 \le \dfrac{x}{3} \\[0.5em] \Rightarrow x \ge -3

Solving right side:

8x3+2143+2x8x32x14328x6x314632x3832x8x4\dfrac{8x}{3} + 2 \le \dfrac{14}{3}+2x \\[0.5em] \Rightarrow \dfrac{8x}{3} - 2x \le \dfrac{14}{3} -2 \\[0.5em] \Rightarrow \dfrac{8x-6x}{3} \le \dfrac{14-6}{3} \\[0.5em] \Rightarrow \dfrac{2x}{3} \le \dfrac{8}{3} \\[0.5em] \Rightarrow 2x \le 8 \\[0.5em] \Rightarrow x \le 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.

Solve the inequation and represent the solution set on the number line: -3 + x ≤ (8x/3) + 2 ≤ (14/3) + 2x, where x ∈ I. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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>3x2x3x>7x>7x<7Value of x which is greatest = 6\text{Let the greatest integer} = x \\[0.5em] \text{According to the condition}, \\[0.5em] 2x + 7 \gt 3x \\[0.5em] \Rightarrow 2x - 3x \gt - 7 \\[0.5em] \Rightarrow -x \gt - 7 \\[0.5em] \Rightarrow x \lt 7 \\[0.5em] \therefore \text{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 = x3\dfrac{x}{3}

Length of pole which is in the water = x6\dfrac{x}{6}

Given,

x[x3+x6]3x(2x+x6)3x(3x6)3xx23x23x6x-\Big[\dfrac{x}{3} + \dfrac{x}{6}\Big] \ge 3 \\[0.5em] \Rightarrow x - \Big(\dfrac{2x + x}{6}\Big) \ge 3 \\[0.5em] \Rightarrow x - \Big(\dfrac{3x}{6}\Big) \ge 3 \\[0.5em] \Rightarrow x - \dfrac{x}{2} \ge 3 \\[0.5em] \Rightarrow \dfrac{x}{2} \ge 3 \\[0.5em] \Rightarrow x \ge 6

∴ Length of pole which is shortest = 6 meters.

Multiple Choice Questions

Question 1

If x ∈ {-3, -1, 0, 1, 3, 5}, then the solution set of the inequation 3x – 2 ≤ 8 is

  1. {-3, -1, 1, 3}
  2. {-3, -1, 0, 1, 3}
  3. {-3, -2, -1, 0, 1, 2, 3}
  4. {-3, -2, -1, 0, 1, 2}

Answer

Given,

x ∈ {-3, -1, 0, 1, 3, 5}

3x283x8+23x10x103x3133x - 2 \le 8 \\[0.5em] \Rightarrow 3x \le 8 + 2 \\[0.5em] \Rightarrow 3x \le 10 \\[0.5em] \Rightarrow x \le \dfrac{10}{3} \\[0.5em] \Rightarrow x \le 3\dfrac{1}{3}

Solution set = {-3, -1, 0, 1, 3}

∴ Option 2 is the correct option.

Question 2

If x ∈ W, then the solution set of the inequation 3x + 11 ≥ x + 8 is

  1. {-2, -1, 0, 1, 2, …}
  2. {-1, 0, 1, 2, …}
  3. {0, 1, 2, 3, …}
  4. {x : x ∈ R, x ≥ –32\dfrac{3}{2} }

Answer

x ∈ W

3x+11x+83xx8112x3x323x + 11 \ge x + 8 \\[0.5em] \Rightarrow 3x - x \ge 8 – 11 \\[0.5em] \Rightarrow 2x \ge -3 \\[0.5em] \Rightarrow x \ge -\dfrac{3}{2} \\[0.5em]

Solution set = {0, 1, 2, 3, …}

∴ Option 3 is the correct option.

Question 3

If x ∈ W, then the solution set of the inequation 5 - 4x ≤ 2 - 3x is

  1. {…, -2, -1, 0, 1, 2, 3}
  2. {1, 2, 3}
  3. {0, 1, 2, 3}
  4. {x : x ∈ R, x ≤ 3}

Answer

x ∈ W

54x23x4x+3x25x3x35 - 4x \le 2 - 3x \\[0.5em] \Rightarrow -4x + 3x \le 2 - 5 \\[0.5em] \Rightarrow -x \le -3 \\[0.5em] \Rightarrow x \ge 3

Solution set = {3, 4, 5, ..... }.

∴ No option is correct .

Question 4

If x ∈ I, then the solution set of the inequation 1 < 3x + 5 ≤ 11 is

  1. { -1, 0, 1, 2}
  2. { -2, -1, 0, 1}
  3. { -1, 0, 1}
  4. {x : x ∈ R, -43\dfrac{4}{3} <\lt x \le 2}

Answer

x ∈ I

Given,
1 < 3x + 5 ≤ 11

Solving left side,

1<3x+515<3x4<3x3x>4x>43\Rightarrow 1 \lt 3x + 5 \\[0.5em] \Rightarrow 1 - 5 \lt 3x \\[0.5em] \Rightarrow -4 \lt 3x \\[0.5em] \Rightarrow 3x \gt -4 \\[0.5em] \Rightarrow x \gt -\dfrac{4}{3}

Solving right side,

3x+5113x1153x6x243<x23x + 5 \le 11 \\[0.5em] \Rightarrow 3x \le 11-5 \\[0.5em] \Rightarrow 3x \le 6 \\[0.5em] \Rightarrow x \le 2 \\[1.5em] \therefore -\dfrac{4}{3} \lt x \le 2

Solution set = {-1, 0, 1, 2}.

∴ Option 1 is the correct option.

Question 5

If x ∈ R, the solution set of 6 ≤ -3(2x - 4) < 12 is

  1. {x : x ∈ R, 0 < x ≤ 1}
  2. {x : x ∈ R, 0 ≤ x < 1}
  3. {0, 1}
  4. none of these

Answer

x ∈ R

Given,
6 ≤ -3(2x - 4) < 12

Solving left side,

63(2x4)66x+126126x66x6x6x16 \le -3(2x - 4) \\[0.5em] \Rightarrow 6 \le - 6x + 12 \\[0.5em] \Rightarrow 6-12 \le -6x \\[0.5em] \Rightarrow -6 \le -6x \\[0.5em] \Rightarrow 6x \le 6 \\[0.5em] \Rightarrow x \le 1

Solving right side,

3(2x4)<126x+12<126x<12126x<0x>00<x1-3(2x - 4) \lt 12 \\[0.5em] \Rightarrow -6x + 12 \lt 12 \\[0.5em] \Rightarrow -6x \lt 12-12 \\[0.5em] \Rightarrow -6x \lt 0 \\[0.5em] \Rightarrow x \gt 0 \\[1.5em] \therefore 0 \lt x \le 1

Solution set = {x : x ∈ R, 0 <x\lt x \le 1}

∴ Option 1 is the correct option.

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,

5x23(3x)5x293x5x+3x9+28x11x1185x - 2 \le 3(3 - x) \\[0.5em] \Rightarrow 5x - 2 \le 9 - 3x \\[0.5em] \Rightarrow 5x + 3x \le 9 + 2 \\[0.5em] \Rightarrow 8x \le 11 \\[0.5em] \Rightarrow x \le \dfrac{11}{8}

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.

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. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 2

Solve the inequations : 6x - 5 < 3x + 4, x ∈ I

Answer

Given,

6x5<3x+46x3x<4+53x<9x<36x - 5 \lt 3x + 4 \\[0.5em] \Rightarrow 6x - 3x \lt 4 + 5 \\[0.5em] \Rightarrow 3x \lt 9 \\[0.5em] \Rightarrow x \lt 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+52x+3x2x35x2x2x + 5 \le 2x + 3 \\[0.5em] \Rightarrow x - 2x \le 3 - 5\\[0.5em] \Rightarrow -x \le -2\\[0.5em] \Rightarrow x \ge 2

∴ Solution set = {x : x ∈ R, x ≥ 2 }.

The graph of the solution set is represented by thick line starting from and including 2.

Find the solution set of the inequation x + 5 ≤ 2x + 3 ; x ∈ R. Graph the solution set on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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<32x71<32x and 32x72x<3+1 and 2x732x<4 and 2x4x<2 and x2x<2 or x2-1 \lt 3 - 2x \le 7 \\[0.5em] \Rightarrow -1 \lt 3 - 2x \text{ and } 3 - 2x \le 7 \\[0.5em] \Rightarrow 2x \lt 3 + 1 \text{ and } -2x \le 7 - 3 \\[0.5em] \Rightarrow 2x \lt 4 \text{ and } -2x \le 4 \\[0.5em] \Rightarrow x \lt 2 \text{ and } -x \le 2 \\[0.5em] \Rightarrow x \lt 2 \text{ or } x \ge -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.

If x ∈ R (real numbers) and -1 < 3 - 2x ≤ 7, find solution set and represent it on a number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 5

Solve the inequation :

5x+174(x7+25)135+3x17,xR.\dfrac{5x+1}{7} - 4\Big(\dfrac{x}{7}+ \dfrac{2}{5}\Big) \le 1\dfrac{3}{5}+\dfrac{3x-1}{7}, x ∈ \bold{R}.

Answer

Given,

5x+174(x7+25)135+3x17,xR.\dfrac{5x+1}{7} - 4\Big(\dfrac{x}{7}+ \dfrac{2}{5}\Big) \le 1\dfrac{3}{5}+\dfrac{3x-1}{7}, x ∈ \bold{R}.

Multiplying both sides by 35

25x+54(5x+14)56+15x525x+520x5656+15x55x5151+15x5x15x51+5110x102x10210x515\Rightarrow 25x + 5-4(5x + 14) \le 56 + 15x -5 \\[0.5em] \Rightarrow 25x + 5 - 20x - 56 \le 56 + 15x -5 \\[0.5em] \Rightarrow 5x - 51 \le 51 + 15x \\[0.5em] \Rightarrow 5x - 15x \le 51 + 51 \\[0.5em] \Rightarrow -10x \le 102 \\[0.5em] \Rightarrow x \ge -\dfrac{102}{10} \\[0.5em] \Rightarrow x \ge -\dfrac{51}{5}

∴ Solution set = {x : x ∈ R, x 515\ge -\dfrac{51}{5} }.

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

74x+2 and 4x+2<12\Rightarrow 7 \le - 4x + 2 \text{ and } - 4x + 2 \lt 12 \\[0.5em]

Solving left side,

74x+24x274x5x547 \le - 4x + 2 \\[0.5em] \Rightarrow 4x \le 2-7 \\[0.5em] \Rightarrow 4x \le -5 \\[0.5em] \Rightarrow x \le \dfrac{-5}{4}

Solving right side,

4x+2<124x<1224x<104x>10x>52-4x + 2 \lt 12 \\[0.5em] \Rightarrow -4x \lt 12 - 2 \\[0.5em] \Rightarrow -4x \lt 10 \\[0.5em] \Rightarrow 4x \gt -10 \\[0.5em] \Rightarrow x \gt \dfrac{-5}{2}

∴ Solution set = {x : x ∈ R, 52-\dfrac{5}{2} < x ≤ 54-\dfrac{5}{4}}.

The graph of the inequation is represented by thick black line starting from 52-\dfrac{5}{2} (excluding 52-\dfrac{5}{2}) till 54-\dfrac{5}{4} (including 54-\dfrac{5}{4}).

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. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Question 7

If x ∈ R, solve 2x3x+1x3>25x2x - 3 \ge x+\dfrac{1−x}{3} \gt \dfrac{2}{5}x. Also represent the solution on the number line.

Answer

Given,

2x3x+1x3>25x2x-3 \ge x + \dfrac{1-x}{3} \gt \dfrac{2}{5}x

Solving left side,

2x3x+1x32x33x+1x32x32x+133(2x3)2x+16x92x+16x2x1+94x10x104x522x-3 \ge x + \dfrac{1-x}{3} \\[0.5em] \Rightarrow 2x - 3 \ge \dfrac{3x +1 -x}{3} \\[0.5em] \Rightarrow 2x -3 \ge \dfrac{2x + 1}{3} \\[0.5em] \Rightarrow 3 (2x -3) \ge 2x+1 \\[0.5em] \Rightarrow 6x - 9 \ge 2x + 1 \\[0.5em] \Rightarrow 6x - 2x \ge 1+ 9 \\[0.5em] \Rightarrow 4x \ge 10 \\[0.5em] \Rightarrow x \ge \dfrac{10}{4} \\[0.5em] \Rightarrow x \ge \dfrac{5}{2}

Solving right side,

x+1x3>25x3x+1x3>25x2x+13>25x5(2x+1)>6x10x+5>6x10x6x>54x>5x>54x + \dfrac{1-x}{3} \gt \dfrac{2}{5}x \\[0.5em] \Rightarrow \dfrac{3x + 1 - x}{3} \gt \dfrac{2}{5}x \\[0.5em] \Rightarrow \dfrac{2x + 1}{3} \gt \dfrac{2}{5}x \\[0.5em] \Rightarrow 5(2x +1) \gt 6x \\[0.5em] \Rightarrow 10x + 5 \gt 6x \\[0.5em] \Rightarrow 10x - 6x \gt -5 \\[0.5em] \Rightarrow 4x \gt -5 \\[0.5em] \Rightarrow x \gt -\dfrac{5}{4}

From left side we get x52x \ge \dfrac{5}{2} and from right side we get x>54x \gt -\dfrac{5}{4}

x52x \ge \dfrac{5}{2}

∴ Solution set = {x : x ∈ R, x 52\ge \dfrac{5}{2} }

The graph of the inequation is represented by thick black line starting from 52\dfrac{5}{2} (including 52\dfrac{5}{2}).

If x ∈ R, solve 2x - 3 ≥ x + (1−x)/(3) > (2x/5). Also represent the solution on the number line. Linear Inequations, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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,

5x6<4x5x4x<6x<65x - 6 \lt 4x \\[0.5em] \Rightarrow 5x - 4x \lt 6 \\[0.5em] \Rightarrow x \lt 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,

13(x+2)15(x)3x3x53235x3x159232x15732x7×1532x35x352x1712\dfrac{1}{3}(x+2) - \dfrac{1}{5}(x) \ge 3 \\[0.5em] \Rightarrow \dfrac{x}{3} - \dfrac{x}{5} \ge 3 - \dfrac{2}{3} \\[0.5em] \Rightarrow \dfrac{5x- 3x}{15} \ge \dfrac{9-2}{3} \\[0.5em] \Rightarrow \dfrac{2x}{15} \ge \dfrac{7}{3} \\[0.5em] \Rightarrow 2x \ge \dfrac{7 \times 15}{3} \\[0.5em] \Rightarrow 2x \ge 35 \\[0.5em] \Rightarrow x \ge \dfrac{35}{2} \\[0.5em] \Rightarrow x \ge 17\dfrac{1}{2}

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

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