critical values

Question

List the critical values of the related function. Then solve the inequality and show work.

 

      5x        <0

   x  +  1

Answer 1

A critical value is x = -1 as that will yield 0 in the denominator, causing the left-hand side to be undefined. Another critical value is x = 0 as that will cause the left-hand side to equal 0. We consider values of x such that x < -1, -1 < x < 0, and x > 0.

x < -1:

Choose x = -2 such that 5x/(x + 1) = 5(-2)/(-2 + 1) = 10. The expression will be positive when x < -1, so the interval (-∞, -1) is not part of the solution.

-1 < x < 0:

Choose x = -0.2 such that 5x/(x + 1) = 5(-0.2)/(-0.2 + 1) = -1.25. The expression will be negative when -1 < x < 0, so the interval (-1, 0) is part of the solution.

x > 0:

Choose x = 4 such that 5x/(x + 1) = 5(4)/(4 + 1) = 4. The expression will be positive when x > 0, so the interval (0, ∞) is not part of the solution.

Thus, the interval for which 5x/(x + 1) < 0 is simply (-1, 0).