Question 1151023: If (x+1)(x-2) is positive, which statement must be true?
A. x<-1 or x>2
B. x>-1 or x<2
C. -1
D. -2
Found 3 solutions by MathLover1, jim_thompson5910, greenestamps:
Answer by MathLover1(20849)   (Show Source):
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website!
A shorthand way of saying "(x+1)(x-2) is positive" is to write (x+1)(x-2) > 0
Change the inequality sign to an equal sign. Then solve for x using the zero product property.
(x+1)(x-2) = 0
x+1 = 0 or x-2 = 0
x = -1 or x = 2
If x = -1 or x = 2, then (x+1)(x-2) is equal to zero.
Draw a number line. Plot -1 and 2 on the number line. This drawing is optional, but it might help you see the three distinct regions.
Region A in red represents everything to the left of -1.
Let's pick one value from this region, say x = -2
(x+1)(x-2) > 0
(-2+1)(-2-2) > 0 ... plug in x = -2
(-1)(-4) > 0
4 > 0
The last inequality is true, so any value less than -1 will make (x+1)(x-2) > 0 true.
Move onto region B in blue. This is the set of numbers between -1 and 2. Pick something from this region, say x = 0, and plug it in to get
(x+1)(x-2) > 0
(0+1)(0-2) > 0
(1)(-2) > 0
-2 > 0
The last inequality is false, so any value between -1 and 2 will make (x+1)(x-2) > 0 false.
Finally let's check region C in green. Pick a value to the right of 2. I'll pick x = 3.
(x+1)(x-2) > 0
(3+1)(3-2) > 0
(4)(1) > 0
4 > 0
The last inequality is true, so any value larger than 2 will make (x+1)(x-2) > 0 true.
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So,
- if x < -1, then (x+1)(x-2) > 0 is true.
- if -1 < x < 2, then (x+1)(x-2) > 0 is false.
- if x > 2, then (x+1)(x-2) > 0 is true.
This is why the final answer is A. x < -1 or x > 2
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
(x+1)(x-2) is positive if (a) both factors are positive, or (b) both factors are negative.
The product is zero only at x = -1 and x = 2.
Both factors are positive only when x > 2.
Both factors are negative only when x < -1.
ANSWER: x < -1 OR x > 2. Answer choice A.
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