multiply both sides of the equation by x * (x+1) to get:
simplify to get:
combine like terms to get:
subtract 6x and subtract 6 from both sides of the equation to get:
factor to get:
set the equation equal to 0 to get:
find the points where the equation crosses the x-axis.
those points are (-3,0) and (2,0)
your solution will be when the equation of
since you know the points where that equation is zero, and since you know that when the equation is positive, it can only get negative by passing through 0, then you need to test on each side of the zero crossing points to find out when the equation is positvive and when the equation is negative.
pick an x < -3.
solve the equation at that point.
for example:
f(-5) = (-5)^2 + (-5) - 6 which is equal to 25 - 5 - 6 which is equal to 25 - 11 which is equal to 14 which is greater than zero.
the equation is greater than 0 when x is smaller than -3.
pick an x > 2.
solve the equation at that point.
for example:
f(5) = (5)^2 + 5 - 6 which is equal to 25 + 5 - 6 which is equal to 25 - 1 which is equal to 24 which is greater than zero.
the equation is greater than 0 when x is larger than 2.
since you know the equation crossed 0 at x = -3and x = 2, this means that it had to be negative between -3 and 2.
you can test just to make sure by picking an x > -3 and < 2.
for example:
f(-.5) = ((-.5)^2 + 5*(-.5) - 6 which is equal to -6.25 which is less than zero.
the equation is less than 0 when x is greater than -3 and less than 2.
since was derived from , this means that the original equation is true when x is greater than or equal to -3 and less than or equal to 2.
the graph of is shown below. you can see where the graph crosses from positive to negative and then from negative to positive.
the graph of with the graph of is shown below.
the intersection points are marked. The x-coordinate of these points are the same x-coordinates of the points where the graph of crossed the x-axis. those points are at x = -3 and x = 2. the graph of is the blue graph and the graph of is the red graph. you can also see where the blue graph is higher than the red graph and vice versa. these are the same regions where the graph of was greater than zero and less than zero.
