Linear Inequations — Assertion-Reason Type Question

If x ≤ -5, x ∈ W

Assertion (A): The above inequation has no solution.

Reason (R): The whole numbers are always positive.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given, inequation :

⇒ x ≤ -5

⇒ x = {-5, -6, -7, -8, ........}

We are looking for whole numbers that are less than or equal to −5.

But since whole numbers start from 0 and go upward, there are no whole numbers ≤ -5.

So, assertion (A) is true.

Whole numbers include 0, which is neither positive nor negative.

So whole numbers are non-negative, not always positive.

So, reason (R) is false.

Thus, Assertion (A) is true, but Reason (R) is false.

Hence, option 1 is the correct option.

If -5 < x and x ≤ 6, x ∈ R

Assertion (A): The above inequation has no solution.

Reason (R): Infinitely many real numbers lie between -5 and 6.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given; -5 < x and x ≤ 6

⇒ -5 < x ≤ 6

Since, x ∈ R

The inequality does have a solution. In fact, it has infinitely many solutions as infinitely many real numbers lie between -5 and 6.

Thus, Assertion (A) is false, but Reason (R) is true.

Hence, option 2 is the correct option.

If -3 ≤ x < $5\dfrac{2}{3}$, x ∈ Z

Assertion (A): x has nine values.

Reason (R): x = 5 is included in the solution set.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given,

⇒ -3 ≤ x < $5\dfrac{2}{3}$

⇒ -3 ≤ x < $\dfrac{17}{3}$

⇒ -3 ≤ x < 5.66.....

Since, x ∈ Z and -3 ≤ x < 5.66.....

⇒ x = {-3, -2, -1, 0, 1, 2, 3, 4, 5}

Thus, x has nine values.

So, assertion (A) is true.

x = 5 is included in the solution set.

So, reason (R) is true but, reason (R) does not clearly explains assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the incorrect reason for Assertion (A).

Hence, option 4 is the correct option.

Given below is the graphical representation of an inequality on number line.

Assertion (A): It represents real numbers lying between -5 and $\dfrac{3}{2}$ but not -5 and $\dfrac{3}{2}$.

Reason (R): The hole represents absence of the number -5.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

If the number is represented by x, then :

From the number line, we get :

$-5 \lt x \le \dfrac{3}{2}$ and x ∈ R.

∴ It represents real numbers lying between -5 and $\dfrac{3}{2}$ but not -5.

So, assertion (A) is false.

The hole on the number line represents absence of the number -5.

So, reason (R) is true.

Thus, Assertion (A) is false, but Reason (R) is true.

Hence, option 2 is the correct option.