Question
Answers

If the positive square root of x is between 3 and 11, then what is the inequality that represents all possible values of x?
(a) 3 < x < 11
(b) 9 < x < 11
(c) 9 < x < 121
(d) x < 3 or x < 11
(e) x < 9 or x < 121

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Hint: We started solving the problem by writing the inequality for the square root of the x. We check whether there are any negative numbers involved in the inequality which may change the signs. We square this inequality to find all the squares i.e., to find all the values of x.

Complete step by step answer:
Given that the positive square root of x lies between 3 and 11. We need to find the inequality that represents all possible values of x.
According to the problem, we have got the inequality $3<\sqrt{x}<11$.
Since 3 and 11 are greater than zero and the signs of the inequality doesn’t change on squaring the inequality.
So, we square each term of inequality $3<\sqrt{x}<11$.
So, we have got the inequality ${{3}^{2}}<{{\left( \sqrt{x} \right)}^{2}}<{{11}^{2}}$ ---(1).
We know that ${{\left( \sqrt{x} \right)}^{2}}=x$ and we use this result in equation (1).
We have got the inequality $9 < x < 121$.
We have found the inequality that represents all possible values of x as $9 < x < 121$.
∴ The inequality that represents all possible values of x is $9 < x < 121$.

So, the correct answer is “Option C”.

Note: We need to make sure that the sign changes are made correctly while squaring the inequality obtained from the problem. We can expect problems that have negative square roots of x and find all possible values of x and also problems that contain both positive and negative square roots of x.