SOLUTION: Solve the quadratic inequality. Write the solution set in interval notation. Show the complete solution. 𝑥(𝑥+1)>112−5𝑥

x*(x+1) > 112 -5x,      ====> (equivalent transformation)  ====>  

x^2 + x > 112 - 5x,     ====> (equivalent transformation)  ====>  

x^2 +x + 5x - 112 > 0,  ====> (equivalent transformation)  ====>  

x^2 + 6x - 112 > 0.


Factor left side. You will get

(x+14)*(x-8) > 0,  or,  equivalently

(x-(-14)*(x-8) > 0.    (1)


1)  If  x < -14  then each factor in the left side of (1) is negative,
    so the product is positive.


2)  If  -14 < x < 8  then the factor (x-(-14)) is positive, while the factor (x-8) in the left side of (1) is negative,
    so the product is negative.


3)  If 8 < x  then each factor in the left side of (1) is positive,
    so the product is positive.


Answer.  The given inequality has  the union of segments  (,}}}] U [,) as the solution set.