Inequalities — Statement I-II Type Questions

Statement 1: Consider the inequation: $3x - 18 \gt 0$. Given replacement set is {0, 1, 2, 3, 4, 5, 6}, then the solution set, is null set i.e. φ.

Statement 2: The solution set of an inequation contains the elements which belong to the replacement set.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Solving,

$\Rightarrow 3x \gt 18$

$\Rightarrow x \gt 6$ [Dividing both sides by 3]

Since there is no element greater than 6 in the replacement set {0, 1, 2, 3, 4, 5, 6}, the solution set is a null set i.e. φ.

∴ Statement 1 is true.

Each element of the solution set belongs to the replacement set.

∴ Statement 2 is true.

Hence, Option 1 is the correct answer.

Statement 1: The solution set of the inequation $2(x - 3) \lt 6 + 6x$ is {-2, -1, .. ..}, where the replacement set is a set of negative integers.

Statement 2: When we multiply both the sides of a linear inequation by a negative number, the direction of inequality symbol changes.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Solving,

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

As the replacement set is the set of negative integers, the solution set is {-1, -2} only and not {-2, -1, .....}.

∴ Statement 1 is false.

On multiplying both the sides of a linear inequation by a negative number, the direction of the inequality symbol changes (reverses).

∴ Statement 2 is true.

Hence, Option 4 is the correct answer.