Linear Inequations

Q1. Multiple Choice Type

i. If 11 + 2x > 5 and x ∈ {negative integers}, then the set to which x belongs is:

Step 1: Solve the inequality.
11 + 2x > 5
Step 2: Subtract 11 from both sides.
2x > -6
Step 3: Divide both sides by 2 (positive).
x > -3
Step 4: Negative integers are {…,-4,-3,-2,-1}.
Greater than -3 means {-2,-1}.
Answer: a. {-2, -1}

ii. If 3x + 1 ≤ 16 and x ∈ {real numbers}, then the values of x represented on a number line are:

Step 1: 3x + 1 ≤ 16
Step 2: 3x ≤ 15
Step 3: x ≤ 5
Step 4: All real numbers to the left of 5 inclusive.

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

Answer: c.

iii. If 3 – 2x ≥ x – 10, x ∈ W, the solution set is:

Step 1: 3 – 2x ≥ x – 10
Step 2: 13 ≥ 3x
Step 3: x ≤ 13/3 ≈ 4.33
Step 4: Whole numbers allowed = {0,1,2,3,4}
Answer: c. {0,1,2,3,4}

iv. If x ∈ W and \(\frac{2x-1}{3}\geq4\), the solution set is:

Step 1: (2x – 1)/3 ≥ 4
Step 2: 2x – 1 ≥ 12
Step 3: 2x ≥ 13
Step 4: x ≥ 6.5 → for whole numbers, x ≥ 7
So solution = {7,8,9,10,…}
Answer: c. {7, 8, 9, 10, …}

v. If x is an integer and -2(x + 3) > 5; the solution set on the number line is:

Step 1: -2(x + 3) > 5
Step 2: x + 3 < -2.5(dividing by -2 reverses inequality)
Step 3: x < -5.5
Step 4: Integers less than -5.5: {…, -7, -6}
On the number line it is all integers to the left of -5.5.

 ←─●──●──|──|──|──|──|──|──|──|──|──|──|──→
  -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5

Answer: d.

Q2. Solve: 7 > 3x – 8 ; x ∈ N

Step 1: Start with the given inequality.
7 > 3x – 8
Step 2: Add 8 to both sides.
7 + 8 > 3x
15 > 3x
Step 3: Divide both sides by 3 (positive).
15 ÷ 3 > x
5 > x
Step 4: This means x < 5.
Since x ∈ N (natural numbers = {1,2,3,4,…}), possible values are {1,2,3,4}.
Answer: {1, 2, 3, 4}

Q3: Solve: -17 < 9y – 8, y ∈ Z

Step 1: Write the given inequality
-17 < 9y – 8
Step 2: Add 8 to both sides
-17 + 8 < 9y
-9 < 9y
Step 3: Divide both sides by 9 (positive number, inequality stays the same)
(-9)/9 < y
-1 < y
Step 4: Since y ∈ Z (integers), solution set is all integers greater than -1
y = 0, 1, 2, 3, …
Answer: {0, 1, 2, 3, …}

Q4: Solve: \(\frac{2}{3} x + 8 < 12, x ∈ W\)

Step 1: Write the given inequality
(2/3)x + 8 < 12
Step 2: Subtract 8 from both sides
(2/3)x < 12 – 8
(2/3)x < 4
Step 3: Multiply both sides by 3/2 (positive number, inequality remains same)
x < 4 × (3/2)
x < 6
Step 4: Since x ∈ W (whole numbers: 0,1,2,…), solution set is
x = 0, 1, 2, 3, 4, 5
Answer: {0, 1, 2, 3, 4, 5}

Q5: Solve the inequation \(8-2x \geq x-5; x \in N\).

Step 1: Write the given inequality \[ 8 – 2x \ge x – 5 \]Step 2: Subtract \(x\) from both sides \[ 8 – 2x – x \ge -5\\ 8 – 3x \ge -5 \]Step 3: Subtract 8 from both sides \[ -3x \ge -13 \]Step 4: Divide both sides by -3 (reverse inequality) \[ x \le \frac{-13}{-3}\\ x \le \frac{13}{3} \approx 4.33 \]Step 5: Since \(x \in \mathbb{N}\), solution set is \[ x = 1, 2, 3, 4 \]Answer: {1, 2, 3, 4}

Q6: Solve the inequality 18 – 3(2x – 5) > 12, x ∈ W

Step 1: Expand the brackets \[ 18 – 3 \cdot 2x + 3 \cdot 5 \gt 12\\ 18 – 6x + 15 \gt 12\\ 33 – 6x \gt 12 \]Step 2: Subtract 33 from both sides \[ -6x \gt 12 – 33\\ -6x \gt -21 \]Step 3: Divide both sides by -6 (reverse inequality because dividing by negative) \[ x \lt \frac{-21}{-6}\\ x \lt 3.5 \]Step 4: Since x ∈ W (whole numbers: 0,1,2,…), solution set is \[ x = 0, 1, 2, 3 \]Answer: {0, 1, 2, 3}

Q7: Solve: 4x – 5 > 10 – x, x ∈ {0,1,2,3,4,5,6,7}

Step 1: Write the given inequality \[ 4x – 5 \gt 10 – x \]Step 2: Add x to both sides \[ 4x + x – 5 \gt 10\\ 5x – 5 \gt 10 \]Step 3: Add 5 to both sides \[ 5x \gt 15 \]Step 4: Divide both sides by 5 \[ x \gt 3 \]Step 5: Since x ∈ {0,1,2,3,4,5,6,7}, the solution set is \[ x = 4, 5, 6, 7 \]Answer: {4, 5, 6, 7}

Q8: Solve: 15 – 2(2x – 1) < 15, x ∈ Z

Step 1: Expand the brackets \[ 15 – 4x + 2 \lt 15\\ 17 – 4x \lt 15 \]Step 2: Subtract 17 from both sides \[ -4x \lt 15 – 17\\ -4x \lt -2 \]Step 3: Divide both sides by -4 (reverse inequality) \[ x \gt \frac{-2}{-4}\\ x \gt \frac{1}{2} \]Step 4: Since x ∈ Z (integers), solution set is \[ x = 1, 2, 3, \dots \]Answer: {1, 2, 3, …}

Q9: Solve: \( \frac{2x + 3}{5} > \frac{4x – 1}{2}, x ∈ W\)

Step 1: Write the given inequality \[ \frac{2x + 3}{5} \gt \frac{4x – 1}{2} \]Step 2: Eliminate denominators by multiplying both sides by 10 (LCM of 5 and 2) \[ 10 \cdot \frac{2x + 3}{5} \gt 10 \cdot \frac{4x – 1}{2}\\ 2 \cdot (2x + 3) \gt 5 \cdot (4x – 1)\\ 4x + 6 \gt 20x – 5 \]Step 3: Subtract 4x from both sides \[ 6 \gt 16x – 5 \]Step 4: Add 5 to both sides \[ 11 \gt 16x\\ 16x < 11 \]Step 5: Divide both sides by 16 \[ x < \frac{11}{16} \]Step 6: Since x ∈ W (whole number), solution set is \[ x = 0 \]Answer: {0}

Q10: 5x + 4 > 8x – 11, x ∈ Z

Step 1: Write the given inequality \[ 5x + 4 \gt 8x – 11 \]Step 2: Subtract 8x from both sides \[ 5x – 8x + 4 \gt -11\\ -3x + 4 \gt -11 \]Step 3: Subtract 4 from both sides \[ -3x \gt -11 – 4\\ -3x \gt -15 \]Step 4: Divide both sides by -3 (reverse inequality) \[ x \lt \frac{-15}{-3}\\ x \lt 5 \]Step 5: Since x ∈ Z (integers), solution set is \[ x = …, -2, -1, 0, 1, 2, 3, 4 \]Answer: {…, -2, -1, 0, 1, 2, 3, 4}

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

Q11: \( \frac{2x}{5} + 1 < -3, x ∈ R\)

Step 1: Write the given inequality \[ \frac{2x}{5} + 1 < -3 \]Step 2: Subtract 1 from both sides \[ \frac{2x}{5} < -3 – 1\\ \frac{2x}{5} < -4 \]Step 3: Multiply both sides by 5 (positive number, inequality remains same) \[ 2x < -20 \]Step 4: Divide both sides by 2 (positive number) \[ x < -10 \]Answer: x ∈ (-∞, -10)

Number line: 
°°°
←───●───●───●───○───|───|───|───|───→
   -13 -12 -11 -10 -9  -8

Q12: \( \frac{x}{2} > -1 + \frac{3x}{4}, x ∈ N\)

Step 1: Write the given inequality \[ \frac{x}{2} > -1 + \frac{3x}{4} \]Step 2: Subtract 3x/4 from both sides \[ \frac{x}{2} – \frac{3x}{4} > -1\\ \frac{2x – 3x}{4} > -1\\ -\frac{x}{4} > -1 \]Step 3: Multiply both sides by -4 (reverse inequality) \[ x < 4 \]Step 4: Since x ∈ N (natural numbers), solution set is \[ x = 1, 2, 3 \]Answer: {1, 2, 3}

Number line: 

←───|───|───●───●───●───|───|───|───→
   -1   0   1   2   3   4   5   6

Q13: \( \frac{2}{3}x + 5 ≤ \frac{1}{2}x + 6, x ∈ W\)

Step 1: Write the given inequality \[ \frac{2}{3}x + 5 \le \frac{1}{2}x + 6 \]Step 2: Subtract (1/2)x from both sides \[ \frac{2}{3}x – \frac{1}{2}x + 5 \le 6\\ \left(\frac{4 – 3}{6}\right)x + 5 \le 6\\ \frac{1}{6}x + 5 \le 6 \]Step 3: Subtract 5 from both sides \[ \frac{1}{6}x \le 1 \]Step 4: Multiply both sides by 6 \[ x \le 6 \]Step 5: Since x ∈ W (whole numbers: 0,1,2,…), solution set is \[ x = 0, 1, 2, 3, 4, 5, 6 \]Answer: {0, 1, 2, 3, 4, 5, 6}

Number line: 

←───|───●───●───●───●───●───●───●───|───|───→
   -1   0   1   2   3   4   5   6   7   8

Q14: Solve the inequation \(5\left(x-2\right) > 4\left(x+3\right)-24\) and represent its solution on a number ine. Given the replacement set is {-4, -3, -2, -1, 0, 1, 2, 3, 4}

Step 1: Expand both sides \[ 5x – 10 \gt 4x + 12 – 24\\ 5x – 10 \gt 4x – 12 \]Step 2: Subtract 4x from both sides \[ 5x – 4x – 10 \gt -12\\ x – 10 \gt -12 \]Step 3: Add 10 to both sides \[ x \gt -2 \]Step 4: Given x ∈ {-4, -3, -2, -1, 0, 1, 2, 3, 4}, select all numbers greater than -2 \[ x = -1, 0, 1, 2, 3, 4 \]Answer: {-1, 0, 1, 2, 3, 4}

Number line: 

←───|───|───|───●───●───●───●───●───●───|───|───→
   -4  -3  -2  -1   0   1   2   3   4   5   6

Q15: Solve \(\frac{2}{3}\left(x-1\right)+4 < 10\) and represent its solution on a number line. Given replacement set is {-8, -6, -4, 3, 6, 8, 12}.

Step 1: Write the inequality \[ \frac{2}{3}(x – 1) + 4 < 10 \]Step 2: Subtract 4 from both sides \[ \frac{2}{3}(x – 1) < 6 \]Step 3: Multiply both sides by 3/2 (positive number) \[ x – 1 < 6 \cdot \frac{3}{2}\\ x – 1 < 9 \]Step 4: Add 1 to both sides \[ x < 10 \]Step 5: Select values from the replacement set {-8, -6, -4, 3, 6, 8, 12} that satisfy x < 10 \[ x = -8, -6, -4, 3, 6, 8 \]Answer: {-8, -6, -4, 3, 6, 8}

Number line: 

←──●─|─●─|─●─|─|─|─|─|─|─●─|─|─●─|─●─|─|─|─|──→
  -8  -6  -4  -2   0   2   4   6   8  10   12