Solve   2x-1 < (x+7)/(x+1)  algebraically

Contrary to what the other person states, you CAN multiply by the LCD, x + 1

Since denominator is x + 1, it follows that: 
          
(2x - 1)(x + 1) < x + 7 ----- Multiplying by LCD, x + 1
    
(x - 2)(x + 2) < 0
x - 2 < 0         OR      x + 2 < 0
x < 2             OR      x < - 2
Since , - 2, - 1, and 2 will be the CRITICAL points.

With the 3 CRITICAL POINTS above, we get the 4 TEST INTERVALS: 
  x < - 2___x = - 3    - 2 < x < - 1___x = - 1.5  -1 < x < 2__x = 0    x > 2___x = 3 
       

Solution sets: x < - 2, and - 1 < x < 2 since these are the ONLY TRUE STATEMENTS.
In INTERVAL NOTATION, we get: ()()