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Homework Help: Range of value to satisfy sufficient and necessary condition

  1. Nov 1, 2015 #1
    1. The problem statement, all variables and given/known data
    p: x and y satisfy inequality x2 + y2 + 4x - 8y + c < 0 where c is real number
    q: x and y satisfy x - y + 8 > 0 ; 4x + 3y - 24 < 0 ; y < 0

    Find the range of c so that:
    a. p is sufficient condition for q
    b. p is necessary condition for q

    2. Relevant equations

    3. The attempt at a solution
    p is the region inside circle (excluding circumference), centered at (-2, 4) and has radius of √(20 - c)

    q is the infinite region bounded by two lines and x - axis

    I want to ask whether my opinion is correct:
    a. For p to be sufficient condition for q, region q should be located inside the circle
    b. For p to be necessary condition for q, the circle should be inside the region of q

  2. jcsd
  3. Nov 1, 2015 #2


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    I think it's the other way around.
    'p is sufficient for q' means ##p\Rightarrow q## which should mean that ##p\subseteq q##.
  4. Nov 1, 2015 #3


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    'p' is the interior of a circle with radius depending on c, 'q' is the region between two intersecting lines and below the y-axis . 'if p then q' (p is a sufficient condition for q) says that if (x,y) is in the circle it must be in the region between the lines- the circle must be a subset of the region. 'if q then p' (p is a necessary condition for q) says that if (x, y) is in the region it must be a in the circle- the region is a subset of the circle. But since the region between the line extends infinitely, that does not see, possible for any 'c'. If the last condition were "y> 0", then it would be a bounded triangular region.
  5. Nov 1, 2015 #4
    I get it.

    Thanks a lot for all the help
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