Solve 5x– 3 < 7, when

Question:
Solve $5 x-3<7$, when

(i) $x$ is an integer

(ii) $x$ is a real number

Solution:

The given inequality is $5 x-3<7$.

$5 x-3<7$

$\Rightarrow 5 x-3+3<7+3$

$\Rightarrow 5 x<10$

$\Rightarrow \frac{5 x}{5}<\frac{10}{5}$

$\Rightarrow x<2$

(i) The integers less than 2 are $\ldots,-4,-3,-2,-1,0,1$.

Thus, when x is an integer, the solutions of the given inequality are

…, –4, –3, –2, –1, 0, 1.

Hence, in this case, the solution set is $\{\ldots,-4,-3,-2,-1,0,1\}$.

(ii) When $x$ is a real number, the solutions of the given inequality are given by $x<2$, that is, all real numbers $x$ which are less than 2 .

Thus, the solution set of the given inequality is $x \in(-\infty, 2)$.