Finding the Interval for -2(x-1)(2x+5)(x-7) > 0

[jeahomgrajan]

Homework Statement

determine the interval algebraically for -2(x-1)(2x+5)(x-7)>0

The Attempt at a Solution

the zeroes are 1. -2.5, 7

what do i do with the -2 in front?

 

[Mentallic]

If you graph $$y=(x-1)(2x+5)(x-7)$$ and then multiply the function by 2, so now you have the graph $$y=2(x-1)(2x+5)(x-7)$$ and finally, take the negative.

Now do you know what $$y=-2(x-1)(2x+5)(x-7)$$ looks like?

 

[jeahomgrajan]

i think I've figured it out, so basically i would divide both sides by -2, and when i divide and an equality by a -, i will hav eot change the direction of the sign

 

[Mentallic]

Yes you can do that too. But understanding what happens to the function when you multiply it by a negative constant would be a useful tool to have in your arsenal

 

[HallsofIvy]

As you say, the zeroes are -2.5, 1, and 7. Suppose x< -2.5. Then each of the factors is negative. Since there are 4 factors, counting the '-2', there product, and the function value, is positive. Now take x between -2.5 and 1. The single factor (2x+ 5)= -2(x- 2.5) changes sign so there are now three negative factors: the function value is negative between -2.5 and 1. Take x between 1 and 7. Now the x-1 factor changes sign to postive and there are now 2 negative factors: the function value is positive between 1 and 7. Finally, if x> 7, all factors except the '-2' are positive: for x> 7, the function value is negative.