Linear Inequations — Multiple Choice Questions

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}

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

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

∴ Option 2 is the correct option.

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 ≥ –$\dfrac{3}{2}$ }

Answer

x ∈ W

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

Solution set = {0, 1, 2, 3, …}

∴ Option 3 is the correct option.

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

$\Rightarrow 5 - 4x \ge 2 - 3x \\[0.5em] \Rightarrow -3x + 4x \le 5 - 2 \\[0.5em] \Rightarrow x \le 3.$

Since, x ∈ W

Solution set = {0, 1, 2, 3}.

∴ Option 3 is the correct option.

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, -$\dfrac{4}{3}$ $\lt$ x $\le$ 2}

Answer

x ∈ I

Given,
1 < 3x + 5 ≤ 11

Solving left side,

$\Rightarrow 1 \lt 3x + 5 \\[0.5em] \Rightarrow 1 - 5 \lt 3x \\[0.5em] \Rightarrow -4 \lt 3x \\[0.5em] \Rightarrow 3x \gt -4 \\[0.5em] \Rightarrow x \gt -\dfrac{4}{3}$

Solving right side,

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

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

∴ Option 1 is the correct option.

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,

$6 \le -3(2x - 4) \\[0.5em] \Rightarrow 6 \le - 6x + 12 \\[0.5em] \Rightarrow 6-12 \le -6x \\[0.5em] \Rightarrow -6 \le -6x \\[0.5em] \Rightarrow 6x \le 6 \\[0.5em] \Rightarrow x \le 1$

Solving right side,

$-3(2x - 4) \lt 12 \\[0.5em] \Rightarrow -6x + 12 \lt 12 \\[0.5em] \Rightarrow -6x \lt 12-12 \\[0.5em] \Rightarrow -6x \lt 0 \\[0.5em] \Rightarrow x \gt 0 \\[1.5em] \therefore 0 \lt x \le 1$

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

∴ Option 1 is the correct option.

The solution set for the inequation 2x + 4 ≤ 14, x ∈ W is :

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

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

3. {1, 2, 3, 4}

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

Answer

Solving,

⇒ 2x + 4 ≤ 14

⇒ 2x ≤ 14 - 4

⇒ 2x ≤ 10

⇒ x ≤ $\dfrac{10}{2}$

⇒ x ≤ 5.

Since, x ∈ W,

∴ x = {0, 1, 2, 3, 4, 5}.

Hence, Option 2 is the correct option.

Given, x + 2 ≤ $\dfrac{x}{3} + 3$ and x is a prime number. The solution set for x is :

1. ∅

2. {0}

3. {1}

4. {0, 1}

Answer

Solving the given equation :

$\Rightarrow x + 2 \le \dfrac{x}{3} + 3 \\[1em] \Rightarrow x - \dfrac{x}{3} \le 3 - 2 \\[1em] \Rightarrow \dfrac{3x - x}{3} \le 1 \\[1em] \Rightarrow \dfrac{2x}{3} \le 1 \\[1em] \Rightarrow x \le \dfrac{3}{2} \\[1em] \Rightarrow x \le 1.5$

Since, x is a prime number less than 1.5

Solution set is empty.

Hence, Option 1 is the correct option.