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Which of the following inequalities, where x is an integer, is/are satisfied by the solution set x = 3, 4, 5? (I) 3x > 9 (II) x − 1 ≥ 2 (III) x + 3 ≥ 5

Which of the following inequalities, where x is an integer, is/are satisfied by the solution set x = 3, 4, 5? (I) 3x > 9 (II) x − 1 ≥ 2 (III) x + 3 ≥ 5 Correct Answer II and III only

Calculation

Given:

(I) 3x > 9

Divide by 3 in above inequality (Here 3 is a positive number so the direction of the inequality does not change)

⇒ x > 9/3

⇒ x > 3

∴ Solution set is (3, ∞)

It doesn’t satisfied the given solution set x = 3

So, statement 1 is wrong.

(II) x − 1 ≥ 2

⇒ x ≥ 2 + 1

⇒ x ≥ 3

∴ Solution set is [3, ∞)

It satisfied the given solution set.

So, statement 2 is correct.

(III) x + 3 ≥ 5

⇒ x ≥ 5 – 3

⇒ x ≥ 2

∴ Solution set is [2, ∞)

It satisfied the given solution set.

So, statement 3 is correct.

Alternate solution:

Consider the given three inequalities for the solution set x = 3, 4, 5

Therefore, substituting 3 in all the three inequalities

1) 3x > 9

⇒ 9 > 9

So, it is false (9 = 9 and is not greater the 9)

2) x – 1 ≥ 2

⇒ 2 ≥ 2

Hence it is true.

3) x + 3 ≥ 5

⇒ 6 ≥ 5

Hence the inequality is true.

For x = 4, 5 inequality II and III are satisfied

Hence, the answer is option 3.

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