Question

# 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

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.

According to the problem, we have got the inequality $3<\sqrt{x}<11$.
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 $9 < x < 121$.