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1. Write the word statement for each of the following inequations:

(a) x ≥ 8

(b) x < -2

(c) x > 6

(d) x ≤ 5

## Solution:

(a) x ≥ 8

The variable x is greater than equal to 8. The possible values of x are 8 and more than 8.

(b) x < -2

The variable x is less than -2. The possible values of x are -1 and less than it.

(c) x > 6

The variable x is greater than 6. The possible values of x are 7, 8, and so on.

(d) x ≤ 5

The variable x is less than and equal to 5. The possible values of x are 5, 4, 3, 2, 1 and so on.

2. Write the resulting equation in each of the following cases when each side of the equation:

(a) x < 2 increased by 2.

(b) x > 4 decreased by 7.

(c) x ≤ 6 multiplied by 4.

(d) x ≥ 25 divided by 5.

## Solution:

(a) x < 2 increased by 2.

Add 2 to both sides of inequation.

x + 2 < 2 + 2

x + 2 < 4.

(b) x > 4 decreased by 7.

Subtract 7 from both sides of the inequation

x – 7 > 4 – 7

x – 7 > -3

(c) x ≤ 6 multiplied by 4.

x x 4 ≤ 6 x 4

4x ≤ 24

(d) x ≥ 25 divided by 5.

x/5 ≥ 25/5

x/5 ≥ 5

3. Write the word statement for the following linear inequations:

(a) x ≤ -4

(b) x > 9

(c) x < -1

## Solution:

(a) x ≤ -4

The variable x is less than and equal to -4. The possible values of x are -4, -5, -6, and so on.

(b) x > 9

The variable x is greater than 9. The possible values of x are 10, 11, 12, and so on.

(c) x < -1

The variable x is less than -1. The possible values of x are -2, -3, -4, -5, -6, and so on.

4. Write the resulting inequation for each question:

(a) x < 5 multiplied by 2.

(b) x > 6 divided by 3.

(c) x ≥ 10 increased by 5.

## Solution:

(a) x < 5 multiplied by 2.

Multiply 2 by both sides.

x x 2 < 5 x 2

2x < 10.

(b) x > 6 divided by 3.

Divide 3 by each side

x/3 > 6/3

x/3 > 2.

(c) x ≥ 10 increased by 5.

Add 5 to each side

x + 5 ≥ 10 + 5

x + 5 ≥ 15.

5. Draw the separate number line for the ‘following inequations’.

(a) x < -9, x ∈ I

(b) x ≥ 7, x ∈ N

(c) x > 8, x ∈ W

## Solution:

(a) x < -9, x ∈ I

Replacement set = {. . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .}

Solution set for given inequation S = {. . . -12, -11, -10}

(b) x ≥ 7, x ∈ N

Replacement set = {1, 2, 3, 4, 5, . . .}

Solution set for given inequation S = {7, 8, 9, . . . }

(c) x > 8, x ∈ W

Replacement set = {0, 1, 2, 3, 4, 5, 6, . . .}

Solution set for given inequation S = {9, 10, 11, 12, 13, . . }

6. Draw a separate number line for the inequation -5 < x ≤ 5 when (i) x ∈ I, (ii) x ∈ W, (iii) x ∈ N. Write the replacement set and solution set in each case.

## Solution:

Given linear inequation is -5 < x ≤ 5

It has two cases.

Case I: -5 < x

Case II: x ≤ 5

(i) x ∈ I

Replacement set = {. . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .}

Solution set for -5 < x is {-4, -3, -2, -1, 0, . . .} = P

Solution set for x ≤ 5 is {. . . 1, 2, 3, 4, 5} = Q

Therefore, required solution set S = P ∩ Q

= {-4, -3, -2, -1, 0, . . .} ∩ {. . . 1, 2, 3, 4, 5}

= {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

(ii) x ∈ W

Replacement set = {0, 1, 2, 3, 4, 5, 6, . . .}

Solution set for -5 < x is {0, 1, 2, 3, 4, 5, 6, . . .} = P

Solution set for x ≤ 5 is {0, 1, 2, 3, 4, 5} = Q

Therefore, required solution set S = P ∩ Q

= {0, 1, 2, 3, 4, 5}

(iii) x ∈ N

Replacement set = {1, 2, 3, 4, 5, . . .}

Solution set for -5 < x is {1, 2, 3, 4, 5, . . .} = P

Solution set for x ≤ 5 is {1, 2, 3, 4, 5} = Q

Therefore, required solution set S = P ∩ Q

= {1, 2, 3, 4, 5}

7. Solve the following linear inequations.

(a) x – 6 < 4, x ∈ W

(b) 4x + 7 > 15, x ∈ N

(c) x/2 + 5 ≤ 6, x ∈ I

## Solution:

(a) x – 6 < 4, x ∈ W

Add 6 to both sides of the linear inequation.

x – 6 + 6 < 4 + 6

x < 10

Replacement set = {0, 1, 2, 3, 4, 5, 6, . . .}

Solution set for x < 10 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(b) 4x + 7 > 15, x ∈ N

Subtract 7 from both sides of the inequation.

4x + 7 – 7 > 15 – 7

4x > 8

Divide each side of the inequation by 4.

4x/4 > 8/4

x > 2.

Replacement set = {1, 2, 3, 4, 5, . . .}

Solution set for x > 2 is {3, 4, 5, . . .}

(c) x/2 + 5 ≤ 6, x ∈ I

Subtract 5 from both sides of the inequation.

x/2 + 5 – 5 ≤ 6 – 5

x/2 ≤ 1

Multiply each side of the inequation by 2.

x/2 x 2 ≤ 1 x 2

x ≤ 2

Replacement set = {. . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .}

Solution set for x ≤ 2 is {. . . -1, 0, 1, 2}

8. Solve the following inequations and represent them graphically.

(a) -x/3 > 5, x ∈ I

(b) 6x – 7 ≥ 17, x ∈ I

(c) -3x > 12, x ∈ I

## Solution:

(a) -x/3 > 5

Multiply each side of the inequation by -3.

-x/3 x (-3) < 5 x (-3)

x < -15

Replacement set = {. . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .}

Therefore, solution set for x > -15 is {. . . -17, -16}

(b) 6x – 7 ≥ 17

Add 7 to both sides.

6x – 7 + 7 ≥ 17 + 7

6x ≥ 24

Divide each side by 6.

6x/6 ≥ 24/6

x ≥ 4

Replacement set = {. . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .}

Therefore, solution set for x ≥ 4 is {4, 5, 6, 7, . . .}

(c) -3x > 12

Divide each side by -3.

-3x/(-3) > 12/(-3)

x < -4

Replacement set = {. . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .}

Therefore, solution set for x < -4 is {. . . -8, -7, -6, -5}

9. Compute the below mentioned inequations.

(a) x/2 > x/3 + 1, x ∈ W

(b) 3x + 5 < 4x – 6, x ∈ W

(c) x/6 – 1 ≥ 3, x ∈ W

(d) 8x – 9 > 9x + 10, x ∈ W

## Solution:

(a) x/2 > x/3 + 1, x ∈ W

Move variable x to one side and constants to other side.

x/2 – x/3 > 1

L.C.M of 3, 2 is 6.

(3x – 2x) / 6 > 1

x/6 > 1

Divide each side by (6)

x/6 x (6) > 1 x (6)

x > 6

Replacement set = {0, 1, 2, 3, 4, 5, 6, . . .}

Therefore, solution set for x > 6 is {7, 8, 9, . .}

(b) 3x + 5 < 4x – 6, x ∈ W

Move variable x to one side and constants to other side.

5 – 6 < 4x – 3x

-1 < x

Replacement set = {0, 1, 2, 3, 4, 5, 6, . . .}

Therefore, solution set for -1 < x is {0, 1, 2, 3, 4, 5, 6, . . .}

(c) x/6 – 1 ≥ 3, x ∈ W

Add 1 to both sides.

x/6 ≥ 3 + 1

x/6 ≥ 4

Multiply each side by 6.

x/6 x 6 ≥ 4 x 6

x ≥ 24

Replacement set = {0, 1, 2, 3, 4, 5, 6, . . .}

Therefore, solution set for x ≥ 24 is {24, 25, 26, . . .}

(d) 8x – 9 > 9x + 10, x ∈ W

Move variable x to one side and constants to other side.

-9 – 10 > 9x – 8x

-19 > x

Replacement set = {0, 1, 2, 3, 4, 5, 6, . . .}

Therefore, solution set for -19 > x is {0, 1, 2, 3, 4, 5, 6, . . .}

10. Represent the following linear inequations graphically.

(a) 7 – 3x < – 2, x ∈ N

(b) 8x – 5 ≥ 18, x ∈ N

(c) 5x/2 > 1, x ∈ N

## Solution:

(a) 7 – 3x < – 2, x ∈ N

Add 3x to each side

7 – 3x + 3x < -2 + 3x

7 < -2 + 3x

Add 2 to both sides

7 + 2 < -2 + 2 + 3x

9 < 3x

Divide each side by 3.

9/3 < 3x/3

3 < x

Replacement set = {1, 2, 3, 4, 5, . . .}

Therefore, solution set for 3 < x is {4, 5, 6, 7. . }

(b) 8x – 5 ≥ 18, x ∈ N

Add 5 to each side

8x – 5 + 5 ≥ 18 + 5

8x ≥ 23

Divide each side by 8.

8x/8 ≥ 23/8

x ≥ 2.87

Replacement set = {1, 2, 3, 4, 5, . . .}

Therefore, solution set for x ≥ 2.87 is {2.87, 3, 4, 5, . . }

(c) 5x/2 > 1, x ∈ N

Multiply each side by 2.

5x/2 x 2 > 1 x 2

5x > 2

Divide each side by 5.

5x/5 > 2/5

x > 0.4

Replacement set = {1, 2, 3, 4, 5, . . .}

Therefore, solution set for x > 0.4 is {1, 2, 3, 4, 5, . . .}