Given:
The inequality is:
[tex]4x+1<5x\leq 3(x+2)[/tex]
To find:
The integer solutions to the given inequality.
Solution:
We have,
[tex]4x+1<5x\leq 3(x+2)[/tex]
This compound inequality can be written as two separate inequalities [tex]4x+1<5x[/tex] and [tex]5x\leq 3(x+2)[/tex].
Now,
[tex]1<5x-4x[/tex]
[tex]1<x[/tex] ...(i)
And,
[tex]5x\leq 3(x+2)[/tex]
[tex]5x\leq 3(x)+3(2)[/tex]
[tex]5x-3x\leq 6[/tex]
[tex]2x\leq 6[/tex]
Divide both sides by 2.
[tex]x\leq \dfrac{6}{2}[/tex]
[tex]x\leq 3[/tex] ...(ii)
From (i) and (ii), we get
[tex]1<x\leq 3[/tex]
Here, 1 is excluded and 3 is included in the solution set. There two integer values 2 and 3 in [tex]1<x\leq 3[/tex].
Therefore, the integer solution for the given inequality are 2 and 3.
