x(x+3)(x-5)>0

Which of the following inequalities specify values of x that satisfy the inequality above?

Indicate all such inequalities.

A. x < -3

B. -3 < x < 0

C. 0 < x < 5

D. x > 5

x(x+3)(x-5)>0

Which of the following inequalities specify values of x that satisfy the inequality above?

Indicate all such inequalities.

A. x < -3

B. -3 < x < 0

C. 0 < x < 5

D. x > 5

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B & C because x cannot equal to -3 , 0 or 5, because on substitution answer will be zero

- x > 0 or
- x+3 > 0

=> x > -3 or - x-5 > 0

=> x > 5

so answer would be B, and D, respectively.

If you get an equation like a.b.c.d > 0 (or a.b.c.d < 0), all you gotta do is draw a number line. Mark all the points where the terms a, b, c, and d are zero. And mark the alternative regions + and -, and pick the regions you want.

For your question, the roots would be x=0,-3 and 5

- Draw a number line and plot the zeroes like this

- Select a point in any region, and find what the value of the expression will be.

For instance, I chose 8 here. So The value of the expression will be 8(11)(3) = +264. So any number in the region (5, \infty) will give us a positive value for the expression. So mark the region as +ve. - Mark the other regions alternately with + and -

- Now, from the question, you want regions where the value is +ve, so your answer is (-3,0) and (5,\infty)

**NOTE**: One exception for this method is when you have the same term repeated an even number of times, say x(x+3)^{2}(x+5). In that case, the equality does not switch to the opposite sign about the point -3

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In option B, X is also less than 0, Does it make sense?

Yes, I guess. as the results of different values need to be positive [x(x+3)(x-5)>0].

in option B, it is clearly indicating that in that range no matter what number we can choose result would be positive. here is what I mean:

let, x = -2, then (-2) *(-2+3)* (-2-5) = (-2)*(1)*(-7) = 14 > 0.

1 Like