Boolean Algebra – Analyzing Boolean Expression x’y + x(x + y’)

Question

I'm still learning Boolean Algebra so I apologise if the question seems pretty straight forward. After working on this problem I thought the answer was 'x', my working out was:

x'y + x(x + y')

= x'y + xx + xy'

= x'y + x + xy'

= x + xy'

= x

but after runnng this through a logic circuit I realise if x = 0 and y = 1 then the ouput does not equal x.
Can someone help me simplify this expression

Answer 1

Two really useful principles are:

Absorption

$x(x+y)=x$

$x+xy=x$

Reduction

$x(x'+y)=xy$

$x+x'y=x+y$

With those:

$x'y+x(x+y')\overset{Absorption}{=} x'y+x\overset{Reduction}{=} y+x$