The number of the real roots of the equation x + 1 2 + | x - 5 | = 27 4 is ________.

Case-$\mathrm{I}$

$x\le 5$

${\left(x+1\right)}^2-\left(x-5\right)=\frac{27}{4}$

${\left(x+1\right)}^2-\left(x+1\right)-\frac{3}{4}=0$

$x+1=\frac{3}{2},-\frac{1}{2}$

$x=\frac{1}{2},-\frac{3}{2}$

Case-$\mathrm{II}$

$x>5$

$\left(x+1\right)+\left(x-5\right)=\frac{27}{4}$

${\left(x+1\right)}^2+\left(x+1\right)-\frac{51}{4}=0$

$x=\frac{-1\pm \sqrt{52}}{2}$ (rejected as $x>5$)

So, the equation have two real root.