Linear Inequations

Q1: If x ∈ {-3, -2, -1, 0, 1, 2, 3}, find the solution of each of the following inequations:

i. x + 2 < 1

Step 1: Subtract 2 from both sides:
x + 2 – 2 < 1 – 2
x < -1
Step 2: Check the values of x from {-3, -2, -1, 0, 1, 2, 3} that satisfy x < -1:
x = {-3, -2}
Answer: x = -3, -2

ii. 2x – 1 < 4

Step 1: Add 1 to both sides:
2x – 1 + 1 < 4 + 1
2x < 5
Step 2: Divide both sides by 2:
x < 5/2
Step 3: From x ∈ {-3, -2, -1, 0, 1, 2, 3}, x < 5/2 = 2.5:
x = -3, -2, -1, 0, 1, 2
Answer: x = {-3, -2, -1, 0, 1, 2}

iii. (2/3)x < 1

Step 1: Multiply both sides by 3/2:
x < 1 × 3/2
x < 3/2
Step 2: x ∈ {-3, -2, -1, 0, 1, 2, 3}, so x < 3/2 = 1.5:
x = -3, -2, -1, 0, 1
Answer: x = {-3, -2, -1, 0, 1}

iv. 1 – x > 0

Step 1: Subtract 1 from both sides:
1 – x – 1 > 0 – 1
-x > -1
Step 2: Multiply both sides by -1 (reverse inequality):
x < 1
Step 3: x ∈ {-3, -2, -1, 0, 1, 2, 3}, so x < 1:
x = -3, -2, -1, 0
Answer: x = {-3, -2, -1, 0}

v. 3 – 5x < -1

Step 1: Subtract 3 from both sides:
3 – 5x – 3 < -1 – 3
-5x < -4
Step 2: Divide by -5 (reverse inequality):
x > 4/5
Step 3: x ∈ {-3, -2, -1, 0, 1, 2, 3}, so x > 0.8:
x = 1, 2, 3
Answer: x = {1, 2, 3}

vi. 2 – 3x > 1

Step 1: Subtract 2 from both sides:
2 – 3x – 2 > 1 – 2
-3x > -1
Step 2: Divide by -3 (reverse inequality):
x < 1/3
Step 3: x ∈ {-3, -2, -1, 0, 1, 2, 3}, so x < 1/3:
x = -3, -2, -1, 0
Answer: x = {-3, -2, -1, 0}

vii. -6 ≥ 2x – 4

Step 1: Add 4 to both sides:
-6 + 4 ≥ 2x – 4 + 4
-2 ≥ 2x
Step 2: Divide both sides by 2:
-1 ≥ x
or x ≤ -1
Step 3: x ∈ {-3, -2, -1, 0, 1, 2, 3}, so x ≤ -1:
x = -3, -2, -1
Answer: x = {-3, -2, -1}

viii. 3x – 5 ≥ -12

Step 1: Add 5 to both sides:
3x – 5 + 5 ≥ -12 + 5
3x ≥ -7
Step 2: Divide by 3:
x ≥ -7/3
Step 3: x ∈ {-3, -2, -1, 0, 1, 2, 3}, so x ≥ -7/3 ≈ -2.33:
x = -2, -1, 0, 1, 2, 3
Answer: x = {-2, -1, 0, 1, 2, 3}

ix. 14 – 2x < 6

Step 1: Subtract 14 from both sides:
14 – 2x – 14 < 6 – 14
-2x < -8
Step 2: Divide by -2 (reverse inequality):
x > 4
Step 3: x ∈ {-3, -2, -1, 0, 1, 2, 3}, so no value satisfies x > 4
Answer: Φ

Q2: If x ∈ N, find the solution set of each of the following inequations:

i. 3x – 8 < 0

Step 1: Add 8 to both sides:
3x – 8 + 8 < 0 + 8
3x < 8
Step 2: Divide both sides by 3:
x < 8/3
Step 3: x ∈ N, so x < 8/3 ≈ 2.67
x = 1, 2
Answer: x = 1, 2

ii. 7x + 3 ≤ 17

Step 1: Subtract 3 from both sides:
7x + 3 – 3 ≤ 17 – 3
7x ≤ 14
Step 2: Divide both sides by 7:
x ≤ 2
Step 3: x ∈ N, so x = 1, 2
Answer: x = 1, 2

iii. 5 – x > 1

Step 1: Subtract 5 from both sides:
5 – x – 5 > 1 – 5
-x > -4
Step 2: Multiply both sides by -1 (reverse inequality):
x < 4
Step 3: x ∈ N, so x = 1, 2, 3
Answer: x = 1, 2, 3

iv. 1 – 3x > -4

Step 1: Subtract 1 from both sides:
1 – 3x – 1 > -4 – 1
-3x > -5
Step 2: Divide by -3 (reverse inequality):
x < 5/3
Step 3: x ∈ N, so x < 5/3 ≈ 1.67
x = 1
Answer: x = 1

v. 3/2 – x/2 > -1

Step 1: Subtract 3/2 from both sides:
3/2 – x/2 – 3/2 > -1 – 3/2
-x/2 > -5/2
Step 2: Multiply both sides by -2 (reverse inequality):
x < 5
Step 3: x ∈ N, so x = 1, 2, 3, 4
Answer: x = 1, 2, 3, 4

vi. -1/4 ≤ 1/2 – x/3

Step 1: Subtract 1/2 from both sides:
-1/4 – 1/2 ≤ 1/2 – 1/2 – x/3
-3/4 ≤ -x/3
Step 2: Multiply both sides by -3 (reverse inequality):
x ≤ 9/4
Step 3: x ∈ N, so x ≤ 9/4 ≈ 2.25
x = 1, 2
Answer: x = 1, 2

Q3: If x ∈ Z, find the solution of each of the following inequations. Represent each solution set on the number line.

i. 9x – 7 ≤ 25 + 3x

Step 1: Subtract 3x from both sides:
9x – 7 – 3x ≤ 25 + 3x – 3x
6x – 7 ≤ 25
Step 2: Add 7 to both sides:
6x – 7 + 7 ≤ 25 + 7
6x ≤ 32
Step 3: Divide both sides by 6:
x ≤ 32/6 = 16/3 ≈ 5.33
Step 4: x ∈ Z, so x ≤ 5
Answer: x = {…, -1, 0, 1, 2, 3, 4, 5}

Number line: 
°°°
←───|──|──|──|──|──|──|──|──|──|───→
   -3 -2 -1  0  1  2  3  4  5  6

ii. -17 < 9x – 8

Step 1: Add 8 to both sides:
-17 + 8 < 9x – 8 + 8
-9 < 9x
Step 2: Divide both sides by 9:
-1 < x
or x > -1
Step 3: x ∈ Z, so x = 0, 1, 2, …
Answer: x = {0, 1, 2, 3, 4, ….}

Number line: 
                                °°°
←───|──|──|──|──|──|──|──|──|──|───→
   -3 -2 -1  0  1  2  3  4  5  6

iii. -4(x + 5) > 10

Step 1: Divide both sides by -4 (reverse inequality):
x + 5 < 10 / -4
x + 5 < -5/2
Step 2: Subtract 5 from both sides:
x < -5/2 – 5 = -15/2 ≈ -7.5
Step 3: x ∈ Z, so x ≤ -8
Answer: x = {…, -10, -9, -8}

Number line: 
°°°
←───|───|───|───|───|───|───|───|───|───|───→
   -14 -13 -12 -11 -10 -9  -8  -7  -6  -5

iv. 4 – 3x < 13 + x

Step 1: Subtract x from both sides:
4 – 3x – x < 13 + x – x
4 – 4x < 13
Step 2: Subtract 4 from both sides:
-4x < 9
Step 3: Divide by -4 (reverse inequality):
x > -9/4 ≈ -2.25
Step 4: x ∈ Z, so x ≥ -2
Answer: x = {-2, -1, 0, 1, 2, 3, …..}

Number line:
                                °°°
←───|──|──|──|──|──|──|──|──|──|───→
   -4 -3 -2 -1  0  1  2  3  4  5

v. 5 – 4x < 10 – x

Step 1: Subtract 10 from both sides:
5 – 4x – 10 < 10 – x – 10
-4x – 5 < -x
Step 2: Add 4x to both sides:
-4x – 5 + 4x < -x + 4x
-5 < 3x
Step 3: Divide by 3:
-5/3 < x
or x > -5/3 ≈ -1.67
Step 4: x ∈ Z, so x ≥ -1
Answer: x = {-1, 0, 1, 2, 3, 4, ….}

Number line: 
                                °°°
←───|──|──|──|──|──|──|──|──|──|───→
   -4 -3 -2 -1  0  1  2  3  4  5

vi. 10 – 2(1 + 4x) < 20

Step 1: Expand brackets:
10 – 2 – 8x < 20
8 – 8x < 20
Step 2: Subtract 8 from both sides:
-8x < 12
Step 3: Divide by -8 (reverse inequality):
x > -12/8 = -3/2 ≈ -1.5
Step 4: x ∈ Z, so x ≥ -1
Answer: x = {-1, 0, 1, 2, 3, ….}

Number line:
                                °°°
←───|──|──|──|──|──|──|──|──|──|───→
   -4 -3 -2 -1  0  1  2  3  4  5

Q4: Find the solution get of each of the following inequations:

i. 1 – 4x ≥ -1, x ∈ N

Step 1: Subtract 1 from both sides:
1 – 4x – 1 ≥ -1 – 1
-4x ≥ -2
Step 2: Divide by -4 (reverse inequality):
x ≤ 1/2
Step 3: x ∈ N = {1,2,3,…}, but x ≤ 1/2 ⇒ No natural number satisfies
Answer: Φ

ii. -3 ≤ 4x + 1 < 9, x ∈ Z

Step 1: Subtract 1 from all parts:
-3 – 1 ≤ 4x + 1 – 1 < 9 – 1
-4 ≤ 4x < 8
Step 2: Divide all parts by 4:
-1 ≤ x < 2
Step 3: x ∈ Z ⇒ x = -1, 0, 1
Answer: x = -1, 0, 1

iii. 0 < 2x – 5 < 5, x ∈ W

Step 1: Add 5 to all parts:
0 + 5 < 2x – 5 + 5 < 5 + 5
5 < 2x < 10
Step 2: Divide by 2:
5/2 < x < 5
Step 3: x ∈ W = {0,1,2,…}, so x = 3, 4
Answer: x = 3, 4

iv. -3 < x/2 – 1 < 1, x ∈ Z

Step 1: Add 1 to all parts:
-3 + 1 < x/2 – 1 + 1 < 1 + 1
-2 < x/2 < 2
Step 2: Multiply all parts by 2:
-4 < x < 4
Step 3: x ∈ Z ⇒ x = -3, -2, -1, 0, 1, 2, 3
Answer: x = -3, -2, -1, 0, 1, 2, 3

v. -4 < 2x/5 + 1 < -3, x ∈ Z

Step 1: Subtract 1 from all parts:
-4 – 1 < 2x/5 + 1 – 1 < -3 – 1
-5 < 2x/5 < -4
Step 2: Multiply all parts by 5/2:
-25/2 < x < -10
Step 3: x ∈ Z ⇒ x = -12, -11
Answer: x = -12, -11

vi. -1 < 2x/3 + 1 ≤ 5, x ∈ Q

Step 1: Subtract 1 from all parts:
-1 – 1 < 2x/3 + 1 – 1 ≤ 5 – 1
-2 < 2x/3 ≤ 4
Step 2: Multiply all parts by 3/2:
-3 < x ≤ 6
Answer: {x ∈ Q:-3 < x ≤ 6}

vii. -1/4 ≤ 1/2 – x/3 < 2, x ∈ Z

Step 1: Subtract 1/2 from all parts:
-1/4 – 1/2 ≤ 1/2 – x/3 – 1/2 < 2 – 1/2
-3/4 ≤ -x/3 < 3/2
Step 2: Multiply all parts by -3 (reverse inequalities):
9/4 ≥ x > -9/2
or -9/2 < x ≤ 9/4
Step 3: x ∈ Z ⇒ x = -4, -3, -2, -1, 0, 1, 2
Answer: x = -4, -3, -2, -1, 0, 1, 2

viii. 3 + x/4 < 2x/3 + 5, x ∈ Q

Step 1: Subtract 3 from both sides:
x/4 < 2x/3 + 2
Step 2: Subtract 2x/3 from both sides:
x/4 – 2x/3 < 2
(-5x)/12 < 2
Step 3: Multiply both sides by -12/5 (reverse inequality):
x > -24/5
Answer: \({x ∈ Q:x > -4\frac{4}{5}}\)

ix. (3x + 1)/4 ≤ (5x – 2)/3, x ∈ Q

Step 1: Multiply both sides by 12:
3*(3x + 1) ≤ 4*(5x – 2)
9x + 3 ≤ 20x – 8
Step 2: Subtract 9x from both sides:
3 ≤ 11x – 8
Step 3: Add 8 to both sides:
11 ≤ 11x
Step 4: Divide by 11:
1 ≤ x
Answer: {x ∈ Q : x ≥ 1}

x. (1/3)(4x – 1) + 3 ≤ 4 + (2/5)(6x + 2), x ∈ Q

Step 1: Expand both sides:
(4x – 1)/3 + 3 ≤ 4 + (12x + 4)/5
(4x – 1)/3 + 3 ≤ 4 + (12x + 4)/5
Step 2: Simplify constants:
(4x – 1)/3 + 3 = (4x – 1)/3 + 9/3 = (4x + 8)/3
(12x + 4)/5 + 4 = (12x + 4)/5 + 20/5 = (12x + 24)/5
Step 3: Inequation:
(4x + 8)/3 ≤ (12x + 24)/5
Step 4: Multiply both sides by 15:
5*(4x + 8) ≤ 3*(12x + 24)
20x + 40 ≤ 36x + 72
Step 5: Subtract 20x from both sides:
40 ≤ 16x + 72
Step 6: Subtract 72 from both sides:
-32 ≤ 16x
Step 7: Divide by 16:
-2 ≤ x
Answer: {x ∈ Q : x ≥ -2}