First slide

Introduction to sets

Question

The number of integral values of x is $5 x - 1 < (x + 1)^{2} < 7 x - 3,$ is

Moderate

A

0
B

1
C

2
D

3

Solution

We have $(x + 1)^{2} > 5 x - 1$
$\begin{array}{ll} \Rightarrow x^{2} - 3 x + 2 > 0 \\ \Rightarrow (x - 1) (x - 2) > 0 \end{array}$
$\Rightarrow x < 1 or x > 2 (i)$
And $(x + 1)^{2} < 7 x - 3$
$\begin{array}{l} \Rightarrow x^{2} - 5 x + 4 < 0 \\ \Rightarrow (x - 1) (x - 4) < 0 \end{array}$
$\Rightarrow 1 < x < 4 (ii)$
From (i) and (ii), common values are, $x \in (2, 4)$
Therefore, there is only one integral value of x which is 3.

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