The correct answer is: [C]: " f(x) ≤ x² − x − 1 " .
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Explanation:
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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: ≥ .
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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" .
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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]" .
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Try Choice: [B]:
→ " y ≤ −x² − x −1 " ;
→ " 2 ≤ −(0²) − 0 − 1 " ?
→ " 2 ≤ 0 − 0 − 1 " ?
→ " 2 ≤ -1 " ? No! So this is a possible solution.
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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.
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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!
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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 " .
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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" .
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Now, let us try "Choice: [C]" :
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→ " 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 " ;
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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" .
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The correct answer is: [C]: " f(x) ≤ x² − x − 1 " .
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