Question

How do I graph each inequality?

(x-5)(x+7)<0

-5x^2+x+2<0

Answer 1

Open interval (-7, 5)

Explanation:

1. `f(x) = (x - 5)(x +7) < 0`
   The 2 x-intercepts are: -7 and 5
   The graph of f(x) is a upward parabola (a > 0).
   Between the 2 x-intercepts, the graph is below the x-axis, meaning (f(x) < 0).
   The solution set is the open intervals: (-7, 5)
   Graph:

----------------(-7) ++++++++ 0 ++++++(5) -----------------

1. `f(x) = - 5x^2 + x + 2 < 0`
   `D = d^2 = b^2 - 4ac = 1 + 40 = 41` --> `d = +- 6.40`
   There are 2 real roots (2 x-intercepts):
   `x = -b/(2a) +- d/(2a) = 1/5 +- 6.40/2 = 0.5 +- 3.2`
   `x1 = 0.5 + 3.2 = 3.7`
   `x2 = 0.5 - 3.2 = - 2.7`
   The parabola graph is downward (a < 0).
   f(x) < 0 whenever the parabola is below the x-axis.
   The solution set is the 2 open intervals:
   (-infinity, -2.7) and (3.7, + infinity)
   Graph.

++++++++++++++ (-2.7)-------- 0 ----------- 3 +++++++++++