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# Which quadrant, if any, contains no point (x, y) that satisfies the in

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Math Expert
Joined: 02 Sep 2009
Posts: 59674
Which quadrant, if any, contains no point (x, y) that satisfies the in  [#permalink]

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21 Nov 2019, 01:07
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Difficulty:

55% (hard)

Question Stats:

45% (01:45) correct 55% (02:18) wrong based on 40 sessions

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Which quadrant, if any, contains no point (x, y) that satisfies the inequality $$y ≥ (x - 3)^2 - 1$$?

(A) I
(B) II
(C) III
(D) IV
(E) All quadrants contain at least one point that satisfies the given inequality.

Attachment:

1.png [ 18.87 KiB | Viewed 441 times ]

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Re: Which quadrant, if any, contains no point (x, y) that satisfies the in  [#permalink]

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21 Nov 2019, 11:14
Bunuel wrote:

Which quadrant, if any, contains no point (x, y) that satisfies the inequality $$y ≥ (x - 3)^2 - 1$$?

(A) I
(B) II
(C) III
(D) IV
(E) All quadrants contain at least one point that satisfies the given inequality.

Attachment:
1.png

Explanation: y ≥ (x - 3)^2 - 1
Solving above equation, we get y ≥ (x-2)(x-4)
So, Putting y=0, We get two values of x=2,4
by putting x=0, we get y ≥ 8

To find out which area is covered by the graph put the Coordinate (0,0) in the original question y ≥ (x - 3)^2 - 1

We get: 0≥8 Which is False.

So (0,0) does not lie in the area covered by the graph, Therefore the equation covers the area above the line.

Thus 4th Quadrant does not contain any point that satisfies the inequality. ( Rest 3 Quadrants will have a few points that would satisfy the inequality).

Hoping other will reply, so we can have good discussion over this question

Please provide kudos, if you find my explanation good enough
Intern
Joined: 11 Sep 2018
Posts: 11
Re: Which quadrant, if any, contains no point (x, y) that satisfies the in  [#permalink]

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24 Nov 2019, 16:40
1
y>= (x-3)^2 -1

I quadrant -> x +ve, y +ve
Ex: (20,5)
20 >= (5-3)^2-1
20>=3
satisfied. Hence we can conclude that it lies in first quadrant.

II Quadrant -> x -ve, y +ve
Ex: (20, -1)
50>=(-1-3)^2-1
50>= 15
Satisfied.

III Quadrant -> x -ve, y -ve
Ex: (-1,-1)
-1>=(-1-3)^2 -1
-1>=15
Not satisfied. As x is within a square term, it will always be positive, y will remain negative. Hence there will be no points in this quadrant.

IV Quadrant -> x +ve, y -ve
(3,-1)
-1>= (3-3)^2-1
-1>=-1
Satisfied.

Ans: C
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Joined: 11 Sep 2018
Posts: 11
Re: Which quadrant, if any, contains no point (x, y) that satisfies the in  [#permalink]

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24 Nov 2019, 16:43
Bunuel VeritasKarishma chetan2u Can you please suggest any alternate solutions other than assuming numbers?
Math Expert
Joined: 02 Aug 2009
Posts: 8305
Which quadrant, if any, contains no point (x, y) that satisfies the in  [#permalink]

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24 Nov 2019, 19:01
1
Bunuel wrote:

Which quadrant, if any, contains no point (x, y) that satisfies the inequality $$y ≥ (x - 3)^2 - 1$$?

(A) I
(B) II
(C) III
(D) IV
(E) All quadrants contain at least one point that satisfies the given inequality.

Attachment:
1.png

Neferteena, the other way would be to simplify the inequality

$$y ≥ (x - 3)^2 - 1$$....... $$y ≥ (x - 3-1)(x-3+1)$$........ $$y ≥ (x - 4)(x-2)$$

(x-4)(x-2) will always be POSITIVE when x is negative, but in quadrant III y is NEGATIVE.
So can negative be greater than positive....NO

C
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Re: Which quadrant, if any, contains no point (x, y) that satisfies the in  [#permalink]

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24 Nov 2019, 23:40
1
Bunuel wrote:

Which quadrant, if any, contains no point (x, y) that satisfies the inequality $$y ≥ (x - 3)^2 - 1$$?

(A) I
(B) II
(C) III
(D) IV
(E) All quadrants contain at least one point that satisfies the given inequality.

Attachment:
The attachment 1.png is no longer available

The inequality is

$$y ≥ (x - 3)^2 - 1$$
$$y ≥ x^2 - 6x + 8$$

We have a quadratic in x which is a parabola. Since the co-efficient of x^2 is positive, it is upward opening. Also, its roots are 4 and 2.
$$x^2 - 6x + 8 = (x - 4)(x - 2) = 0$$

Now the inequality is $$y ≥ x^2 - 6x + 8$$ so which region does it represent?
Let's try to put in (3, 0) lying inside the parabola.
So this is what its graph will look like:

Attachment:

IMG_7737.jpg [ 1.76 MiB | Viewed 153 times ]

0 ≥ - 1 (true)
So our inequality represents the region inside the parabola. So no point of quadrant III will satisfy it.

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Re: Which quadrant, if any, contains no point (x, y) that satisfies the in   [#permalink] 24 Nov 2019, 23:40
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