Question

Solve for x. Enter your answer in interval notation using grouping symbols. x2 + x < 20

Answer 1

Answer:

x<18

Step-by-step explanation:

2+x<20

Answer 2

Answer:

(-5, 4)

Step-by-step explanation:

x^2 + x < 20 can be rewritten as a quadratic in standard form:  x^2 + x - 20 < 0.  Recognize that the graph of x^2 + x - 20 is that of a parabola that opens up.  Because the y-intercept (0, -20) is below the x-axis, we know for certain tht the graph intersects the x-axis in two places.  Our job is to determine the two horizontal intercepts, which in turn determine the solution set.

x^2 + x - 20 = 0 factors as follows:  (x + 5)(x - 4) = 0, whose roots are -5 and 4.

These two roots form 3 intervals:  (-infinity, -5), (-5, 4) and (4, infinity).   We now must identify on which intervals x^2 + x - 20 is less than 0 (that is, on which intervals the graph is below the x-axis).  Choose three test values for x:  the first could be -6 (which is in the set (-infinity, -5) ); x^2 + x - 20 is + there, and so the graph is above the x-axis and x^2 + x - 20 is positive.  Reject this.  

Next, test x = 0; x^2 + x - 20 is negative there, meaning that x^2 + x - 20 < 0 and that x^2 + x < 20.  Thus, the solution is (-5, 4).  It can be shown that x^2 + x - 20 is positive again for x-values within the interval (4, infinity).

Evaluating