The correct answer is: [C]: " f(x) ≤ x² − x − 1 " .<span> __________________________________________________________ Explanation: __________________________________________________________ Note that the graph has a solid line. This means that the correct equation for the graph contain either: 1) </span>≤ or: 2) ≥ .
We note that all of the answer choices provided have either: ≤ or: <span>≥ . </span>_________________________________________________________ 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:
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<span>" . </span>__________________________________________________________ The correct answer is:[C]: " f(x) ≤ x² − x − 1 " . __________________________________________________________
Since we know that the paper is $1.50 per package and she bought 4 packages of paper, then she spent a total of $6 on the paper. Which leaves ($12.25-$6) $6.25 that she spent on buying 5 pencils. Now the cost per pencil can be derived by dividing $6.25/5, which is $1.25. Each pencil is $1.25.
The equation that can be used is the following where the cost per pencil is represented by x: