Which inequality statement best represents the graph?

The correct answer is:  [C]:  "  f(x)  ≤  x²  − x − 1 " .

Explanation:

Note that the graph has a solid line.  This means that the correct equation for the graph contain either:   1)  ≤  or:  2)  ≥ .

We note that all of the answer choices provided have either:  ≤  or:  ≥  .

Let us start with a point on the graph that is "not shaded" (i.e. that is NOT a solution:  " (0, 2) " ; that is:  when "x = 0" , y = 2" .

Try this for all the answer choices:

Choice:  [A]:  " y  ≥  x² − x  −  1 " ;

→  " 2  ≥  0² − 0  −  1  " ? ;  

→  " 2  ≥  0 − 1 " ? ; 

→   "2  ≥ -1 " ?  Yes! 

So we can rule out "Choice: [A]" . 

Try Choice: [B]:

→  " y ≤ −x² − x −1 " ; 

→  " 2 ≤ −(0²) − 0 − 1 " ? 

→  " 2 ≤  0 − 0 − 1 " ?

→  " 2 ≤  -1 " ?  No!  So this is a possible solution.

Now, let's try Choice:  [C]"

→  " y ≤  x²  − x − 1 " ; 

→  " 2 ≤  0² − 0 − 1 " ? ;

→  " 2 ≤  0 − 1 " ? ; 

→  " 2 ≤  -1 " ? ;   No!  So this is possible answer.

Now, let's try the last answer choice:  [D]:

→  " y ≥  −x²  − x − 1 " ;

→  " 2 ≥  −0² − 0 − 1 " ? ; 

→  " 2 ≥  0 − 0 − 1 " ? ; 

→  " 2 ≥  - 1 " ? ;  No!  So this is a possible answer!

Now, we have narrow our choices to B, C, and D.

Now , among these choices, which of these fits a point with the shaded region?

Let us pick a point:  "(2, 0) "  that, is, when "x = 2, y = 0 " .

Let us start with:  "Choice:  [B] " :

→  " y ≤ −x² − x −1 " ; 

→  " 0  ≤ −(2²) − 2 − 1 " ? 

→  " 0 ≤  -4 − 2 − 1 " ?

→  " 0 ≤  -7 " ?  No!  So we can rule out "Choice B" . 

Now, let us try "Choice:  [C]" :


→  " y ≤  x²  − x − 1 " ; 

→  " 0 ≤  2² − 2 − 1 " ? ;

→  " 0 ≤   4 − 2  − 1 " ? ; 

→  " 0  ≤  1 " ? ;   Yes!  So; this should be the correct

Answer choice:  [C]:  " f(x)  ≤  x²  − x − 1 " ; 

Let us try " Choice:  [D]"— to confirm that "Choice: [D]" is incorrect:

→  " 0  ≥  −(2²)  − 2 − 1 " ;

→  " 0   ≥  −(2²) −  2 − 1 " ? ; 

→  " 0   ≥   − 4  −  2 − 1 " ? ; 

→  " 0   ≥   − 7 " ? ;  No!  So we can rule out: "Choice:  D" .

The correct answer is:  [C]:  "  f(x)  ≤  x²  − x − 1 " .