Solve the polynomial inequality and express in interval notation? x^2-2x-15 < 0 I don't understand why the answer (-3,5) does not include negative infinity or infinity in the answer

Mar 17, 2018

$- 3 < x < 5$

Explanation:

We have: ${x}^{2} - 2 x - 15 < 0$

$R i g h t a r r o w {x}^{2} + 3 x - 5 x - 15 < 0$

$R i g h t a r r o w x \left(x + 3\right) - 5 \left(x + 3\right) < 0$

$R i g h t a r r o w \left(x + 3\right) \left(x - 5\right) < 0$

Now, for a product to be less than zero, either one factor must be less than zero and the other greater than zero, i.e. either one factor is positive and one is negative, or vice versa:

$R i g h t a r r o w x + 3 < 0$ and $x - 5 > 0 R i g h t a r r o w x < - 3$ and $x > 5$

or

$R i g h t a r r o w x - 5 < 0$ and $x + 3 > 0 R i g h t a r r o w x < 5$ and $x > - 3$

The first case is not possible ($x$ cannot be greater than $5$ and less than $- 3$).

Therefore, the solution to the inequality is $- 3 < x < 5$.

"Negative" and "positive" infinity aren't included in the answer because $x$ lies between, and only between, $- 3$ and $5$.