Question

For what values of
`x` is the product
`(x+5)(x-1)` positive? Explain.

Answer 1

The regions where (x+5) (x-1) >0 is x< -5 or x>1

Explanation:

(x+5) (x-1) >0

First find the zeros

(x+5) =0 x-1 =0

x= -5 x=1

We then have to check the regions

x< -5 -5 <x<1 x>1

First x< -5

(x+5) (x-1)

X+5 will be negative ( x-1) will be negative

(-) * (-) = + so the values will be positive

Then -5 <x<1

X+5 will be positive ( x-1) will be negative

( +) * (-) will be negative

First x>1

(x+5) (x-1)

X+5 will be positive ( x-1) will be positive

(+) * (+) = + so the values will be positive

The regions where (x+5) (x-1) >0 is x< -5 or x>1

Answer 2

`x<-5\text{ or } x>1`

Explanation:

We have:

`(x+5)(x-1)`

And we want to find the value of x such that the expression is positive. So, we can write this as the following inequality:

`(x+5)(x-1)>0`

Solve for the inequality. First, we can solve for the zeros like a normal quadratic. So, pretend the inequality is with an equal sign:

`(x+5)(x-1)=0`

Zero Product Property:

`x+5=0\text{ or } x-1=0`

On the left, subtract 5.

On the right, add 1.

So, our zeros are:

`x=-5, 1`

Since our inequality is a greater than, our answer is an "or" inequality with our answer being all the values to the left of our lesser zero and all the values to the right of our greater zero.

So, our solution is:

`x<-5\text{ or } x>1`

And we're done!