Inequalities — Multiple Choice Questions

The solution set for $8 - x \ge 3, x$ ∈ N is

1. {5, 6, 7, 8, 9, .....}

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

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

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

Answer

Solving,

$\Rightarrow 8 - x \ge 3\\[1em] \Rightarrow -x \ge 3 - 8\\[1em] \Rightarrow -x \ge -5\\[1em] [\text{Multiplying by -1 and reversing the symbol}] \\[1em] \Rightarrow x \le 5$

As $x$ ∈ N, the solution set is {1, 2, 3, 4, 5}.

Hence, Option 2 is the correct answer.

$x \lt 3, x$ ∈ W on the number line is:

Answer

For $x \lt 3$ and $x$ ∈ W, the solution set is {0, 1, 2}.

The number line showing thick dots at 0, 1 and 2 only, represents the solution set.

Hence, Option 3 is the correct answer.

Solution set for $\dfrac{2x - 1}{3} \le 4, x$ ∈ W is

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

2. {0, 1, 2, 3, 4, 5, 6}

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

4. {7, 8, 9, 10, .......}

Answer

Solving,

$\Rightarrow \dfrac{2x - 1}{3} \le 4\\[1em] \Rightarrow 2x - 1 \le 12\\[1em] \Rightarrow 2x \le 12 + 1\\[1em] \Rightarrow 2x \le 13\\[1em] \Rightarrow x \le \dfrac{13}{2}\\[1em] \Rightarrow x \le 6\dfrac{1}{2}$

As $x$ ∈ W, the solution set is {0, 1, 2, 3, 4, 5, 6}.

Hence, Option 2 is the correct answer.

Solution set for $19 - 3x \ge x - 2, x$ ∈ Z (integers) is

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

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

3. {....., -7, -6, -5}

4. {6, 7, 8, 9, .....}

Answer

Solving,

$\Rightarrow 19 - 3x \ge x - 2\\[1em] \Rightarrow 19 + 2 \ge x + 3x\\[1em] \Rightarrow 21 \ge 4x\\[1em] \Rightarrow \dfrac{21}{4} \ge x\\[1em] \Rightarrow x \le 5\dfrac{1}{4}$

As $x$ ∈ Z, the integers less than or equal to $5\dfrac{1}{4}$ are {...., -2, -1, 0, 1, 2, 3, 4, 5}.

Hence, Option 1 is the correct answer.

Solution set for $2 - 3(5 - x) \ge -1, x$ ∈ N is

1. {1, 2}

2. {0, 1, 2}

3. {3, 4, 5, 6, ....}

4. {4, 5, 6, 7, .......}

Answer

Solving,

$\Rightarrow 2 - 3(5 - x) \ge -1\\[1em] \Rightarrow 2 - 15 + 3x \ge -1\\[1em] \Rightarrow -13 + 3x \ge -1\\[1em] \Rightarrow 3x \ge -1 + 13\\[1em] \Rightarrow 3x \ge 12\\[1em] \Rightarrow x \ge \dfrac{12}{3}\\[1em] \Rightarrow x \ge 4$

As $x$ ∈ N, the solution set is {4, 5, 6, 7, .......}.

Hence, Option 4 is the correct answer.

Solution set for $1 + \dfrac{x - 1}{2} \ge x, x$ ∈ W is

1. {1, 2, 3, .....}

2. {1}

3. {0, 1}

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

Answer

Solving,

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

As $x$ ∈ W, the solution set is {0, 1}.

Hence, Option 3 is the correct answer.

Solution set for $x + 14 \le 8x - 7, x$ ∈ Z (integers) is

Answer

Solving,

$\Rightarrow x + 14 \le 8x - 7\\[1em] \Rightarrow 14 + 7 \le 8x - x\\[1em] \Rightarrow 21 \le 7x\\[1em] \Rightarrow \dfrac{21}{7} \le x\\[1em] \Rightarrow 3 \le x\\[1em] \Rightarrow x \ge 3$

As $x$ ∈ Z, the solution set is {3, 4, 5, 6, .....}, which is shown by thick dots at 3, 4, 5, 6, ..... with the dark arrow head on the right side of the number line.

Hence, Option 2 is the correct answer.

Answer

Set A is a set of integers greater than equal to -1 and is shown by thick dots on the number line, whereas set B is a set of real numbers less than equal to -3 and is shown by dark arrow on the left side of the number line.

A ∪ B is the set of all the elements which belong either to set A or to set B or to both.

∴ A ∪ B is represented by the dark arrow of set B together with the thick dots of set A.

Hence, Option 2 is the correct answer.

Solution set for, $-1 \le x \lt 4$, and $x$ ∈ R (real numbers) on number line is:

Answer

The given inequation is $-1 \le x \lt 4$, where $x$ ∈ R.

As $x$ ∈ R (real numbers), the solution set contains all the real numbers from -1 to 4, so it is represented by a thick (continuous) line from -1 to 4 on the number line.

Since -1 is included in the solution set (as $-1 \le x$), it is marked with a solid (filled) circle, and since 4 is not included in the solution set (as $x \lt 4$), it is marked with a hollow (open) circle.

Hence, Option 4 is the correct answer.

The set ${x : x \in R, x \ge -5 \text{ and } x \lt 2}$ on the number line is

Answer

The given set is ${x : x \in R, x \ge -5 \text{ and } x \lt 2}$, i.e. $-5 \le x \lt 2$, where $x$ ∈ R.

As $x$ ∈ R (real numbers), the set contains all the real numbers from -5 to 2, so it is represented by a thick (continuous) line from -5 to 2 on the number line.

Since -5 is included in the set (as $x \ge -5$), it is marked with a solid (filled) circle, and since 2 is not included in the set (as $x \lt 2$), it is marked with a hollow (open) circle.

Hence, Option 1 is the correct answer.