```Question 20183
Hello There:

w^2 + 3*w < 18

We recognize that subtracting 18 from both sides will lead to a quadratic expression on the left side of the inequality symbol.  Let's do that.

w^2 + 3*w - 18 < 0

Next, we find the values of w that cause the left side to equal zero because these values are the boundary points between where the quadradic expression is larger or smaller than zero.

w^2 + 3*w - 18 = 0

The left side factors.

(w + 6)*(w - 3) = 0

So, we see that the values of -6 and 3 are the boundary points.

These two values divide the number line into three intervals.  We pick a value within each interval to see if it leads to a correct statement when we substitute it into the original inequality given.  (If one value works within an interval, then all of the values in that interval also work.  This is why we only need to text one value in each interval.

For the interval (negative infinity to -6) we pick -10.

(-10)^2 + 3*(-10) < 18

100 - 30 < 18

70 < 18

This is a false statement, so the numbers in this interval are not solutions.

For the interval (-6, 3) we pick zero.

(0)^2 + 3*(0) < 18

0 < 18

This is true, so the values in this interval are solutions.

For the interval (3, positive infinity) we pick 10.

(10)^2 + 3*(10) < 18

100 + 30 < 18

130 < 18

This is false, so this interval is not a solution.

The solution is:

All values between -6 and 3.

~ Mark

P.S.  This could also be done graphically by entering the expressions:

y1 = x^2 + 3*x

y2 = 18

on a graphing calculator, and then zooming in on the x-values where the bottom of the parabola lies beneath the line y = 18.
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