Linear Inequations — Assertion-Reason Type Questions

Assertion (A) : If 8 < 5(x + 1) - 2 ≤ 18, x ∈ R, then the smallest integer value of x is 0.

Reason (R) : Multiplying each side of an inequation by the same integer does not change inequality.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given,

⇒ 8 < 5(x + 1) - 2 ≤ 18

Solving L.H.S of inequation,

⇒ 8 < 5(x + 1) - 2

⇒ 5(x + 1) - 2 > 8

⇒ 5x + 5 - 2 > 8

⇒ 5x + 3 > 8

⇒ 5x > 8 - 3

⇒ 5x > 5

⇒ x > $\dfrac{5}{5}$

⇒ x > 1 ..........(1)

Solving R.H.S of inequation,

⇒ 5(x + 1) -2 ≤ 18

⇒ 5x + 5 - 2 ≤ 18

⇒ 5x + 3 ≤ 18

⇒ 5x ≤ 18 - 3

⇒ 5x ≤ 15

⇒ x ≤ $\dfrac{15}{5}$

⇒ x ≤ 3 ..........(2)

From (1) and (2), we get :

⇒ 1 < x ≤ 3, x ∈ R

∴ The smallest integer value of x is 2.

∴ Assertion (A) is false.

Multiplying each side of an inequation by the same positive integer does not change inequality, while the inequality changes if multiplied by negative integer.

∴ Reason (R) is false.

Hence, Both A and R are false.

Assertion (A) : For the inequation -12 < 3 - 4x ≤ 11, x ∈ N, the solution set is {1, 2, 3, 4}.

Reason (R) : The set of all those values of x from the replacement set which satisfy the given inequation is called the solution set of inequation.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given,

⇒ -12 < 3 - 4x ≤ 11

Solving L.H.S of inequation,

⇒ -12 < 3 - 4x

⇒ 4x < 3 + 12

⇒ 4x < 15

⇒ x < $\dfrac{15}{4}$

⇒ x < 3.75 ........(1)

Solving R.H.S of inequation,

⇒ 3 - 4x ≤ 11

⇒ 11 ≥ 3 - 4x

⇒ 4x ≥ 3 - 11

⇒ 4x ≥ -8

Dividing by 4 on both sides we get,

⇒ x ≥ -2 ..........(2)

From (1) and (2) we get,

⇒ -2 ≤ x < 3.75

Since x ∈ N

Solution set = {1, 2, 3}

∴ Assertion (A) is false.

We know that,

The set of all those values of x from the replacement set which satisfy the given in equation is called the solution set of inequation.

∴ Reason (R) is true.

Hence, Option 4 is the correct option.

Assertion (A) : If 2x - 5 ≤ 5x + 4 < 11, x ∈ I, then greatest value of x is 1.

Reason (R) : Adding or subtracting a negative integer to each side of an inequation reverses the inequality.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given,

⇒ 2x - 5 ≤ 5x + 4 < 11

Solving L.H.S of inequation,

⇒ 2x - 5 ≤ 5x + 4

⇒ 5x + 4 ≥ 2x - 5

⇒ 5x - 2x ≥ -5 - 4

⇒ 3x ≥ -9

⇒ x ≥ $-\dfrac{9}{3}$

⇒ x ≥ -3

Solving R.H.S of inequation,

⇒ 5x + 4 < 11

⇒ 5x < 11 - 4

⇒ 5x < 7

⇒ x < $\dfrac{7}{5}$

⇒ x < 1.4

From (1) and (2) we get,

⇒ -3 ≤ x < 1.4

Since x ∈ I

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

The greatest value of x is 1.

∴ Assertion (A) is true.

We know that,

Adding or subtracting a negative integer to each side of an inequation does not change the inequality.

∴ Reason (R) is false.

Hence, Option 3 is the correct option.