Math question under algebra section.

Question

Show that for all real values of x, (x²+x+1)/(x+1) does not lie between -3 and 1.

Answer 1

Let

 

(x² + x + 1)/(x + 1) = k, 

 

and see if there is a restriction on k.

 

(x² + x + 1) = k(x + 1) = kx + k

 

x² + (1 - k)x + (1 - k) = 0

 

For x to be real, the discriminant of this equation must be ≥ 0.

 

D = (1 - k)² - 4(1 - k) = 1 - 2k + k² - 4 + 4k = k² + 2k - 3 

 

D = (k + 3)(k - 1)

 

D ≥ 0 ⇒ three possibilities:

 

(i)

 

Both factors are positive.  Then k > -3 and k > 1.  The intersection is k > 1.

 

(ii)

 

Both factors are negative.  Then k < -3 and k < 1.   The intersection is k < -3. 

 

(iii)  

 

At least one factor is zero.  Then k = -3 or 1.

 

It looks like the restrictions on k are k ≥ 1 or k ≤ -3.  Combine these two and you have k is not in (-3,1).  Since 

 

k = (x² + x + 1)/(x + 1), then (x² + x + 1)/(x + 1) is not in (-3,1).  But I can not see any problem with the expression being equal to the endpoints of the interval.