Question 381601: Which of the following solutions to the quadratic inequality
(x - 2)(x + 1) > 4
is correct? Explain why the others are incorrect.
a) Solution 1
Since (x - 2)(x + 1) > 4 means that x - 2 > 4 or x + 1 > 4 ,we have the following:
if x - 2 > 4 , then x > 6 and if x + 1 > 4 , then x > 3.
Therefore the solution to the inequality is the interval (3,∞).
b) Solution 2
Since (x - 2)(x + 1) > 4 means that (x - 2)(x + 1) - 4 > 0 , we have the following:
Either (x - 2)(x + 1) > 0 or -4 > 0 and since -4 is never greater than 0 we
need only worry about (x - 2)(x + 1) > 0 .
Now, since (x - 2)(x + 1) > 0 implies that either x - 2 > 0 or x + 1 > 0 , the
solution to the inequality must be the interval (-1,∞).
c) Solution 3
Since (x - 2)(x + 1) > 4 means that (x - 2)(x + 1) - 4 > 0 , and (x - 2)(x + 1) - 4 > 0 is equivalent to writing x^2 - x - 6 > 0 and x^2 - x - 6 = 0 when x = -2 and x = 3 ,the solution to the inequality must be the interval (-2,3).
d) Solution 4
Since (x - 2)(x + 1) > 4 means that (x - 2)(x + 1) - 4 > 0 , and (x - 2)(x + 1) - 4 > 0 is equivalent to writing x^2 - x - 6 > 0 and x^2 - x - 6 = 0 when x = -2 and x = 3 , the solution to the inequality must be the interval (-∞,-2)and(3,∞)
Found 2 solutions by rapaljer, stanbon:
Answer by rapaljer(4671)   (Show Source):
Answer by stanbon(75887)  (Show Source):
You can put this solution on YOUR website! Which of the following solutions to the quadratic inequality
(x - 2)(x + 1) > 4
is correct? Explain why the others are incorrect.
a) Solution 1
Since (x - 2)(x + 1) > 4 means that x - 2 > 4 or x + 1 > 4 ,we have the following:
if x - 2 > 4 , then x > 6 and if x + 1 > 4 , then x > 3.
Therefore the solution to the inequality is the interval (3,∞).
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But x= -3 is also a solution: and x=-4, or x= -5, etc.
And they are not in (3,+inf)
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b) Solution 2
Since (x - 2)(x + 1) > 4 means that (x - 2)(x + 1) - 4 > 0 , we have the following:
Either (x - 2)(x + 1) > 0 or -4 > 0 and since -4 is never greater than 0 we
need only worry about (x - 2)(x + 1) > 0 .
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No: (x-2)(x+1) >4, not just > 0.
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Now, since (x - 2)(x + 1) > 0 implies that either x - 2 > 0 or x + 1 > 0 , the
solution to the inequality must be the interval (-1,∞).
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c) Solution 3
Since (x - 2)(x + 1) > 4 means that (x - 2)(x + 1) - 4 > 0 , and (x - 2)(x + 1) - 4 > 0 is equivalent to writing x^2 - x - 6 > 0 and x^2 - x - 6 = 0 when x = -2 and x = 3 ,the solution to the inequality must be the interval (-2,3).
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No. -2 and 3 are the boundary values of the solution set of the inequality
The solution sets are to the left of -2 and the right of 3.
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d) Solution 4
Since (x - 2)(x + 1) > 4 means that (x - 2)(x + 1) - 4 > 0 , and (x - 2)(x + 1) - 4 > 0 is equivalent to writing x^2 - x - 6 > 0 and x^2 - x - 6 = 0 when x = -2 and x = 3 , the solution to the inequality must be the interval (-∞,-2)and(3,∞)
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True
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Cheers,
Stan H.
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