what are all intervals on which (x-1)/(x^2-16) is positive?

Answer 1

Set the function equal to 0 and undefined and solve for x. At these values of x, the function can change from positive to negative or vice versa.

Setting the function equal to 0 gives x = 1

Setting the function to be undefined, gives x = -4 and 4.

Create a number line

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-4 1 4

On the interval (-∞, -4) the function is either negative or positive

Same for the intervals (-4, 1), (1, 4), and (4, ∞). Plug in a number in each of these intervals and see if the y value is positive or negative.

For example, if we plug in 0 into the function, we get 16, which is positive. That means that the entire interval (-4, 1) is positive