![\"\"](\"/images/stars/stars-13.gif\")
You can put this solution on YOUR website!
Given:
\n" ); document.write( ".
\n" ); document.write( "f(x)=x(x+3)(x-1)
\n" ); document.write( ".
\n" ); document.write( "Suppose we replace f(x) by y so that the equation is a little more in the form of the way
\n" ); document.write( "we are used to thinking about graphs. If we do that the equation becomes:
\n" ); document.write( ".
\n" ); document.write( "y = x(x+3)(x-1)
\n" ); document.write( ".
\n" ); document.write( "Now suppose that a factor on the right side has a value of x such that the factor itself equals
\n" ); document.write( "zero. Having a factor equal to zero on the right side means that the entire right side becomes
\n" ); document.write( "zero because it involves a multiplication by zero. And if the right side equals zero, then
\n" ); document.write( "y equals zero.
\n" ); document.write( ".
\n" ); document.write( "Now think to yourself, \"If a point on a graph has a y value equal to zero, where must that
\n" ); document.write( "point be on the graph?\" With a little thought you will convince yourself that the point
\n" ); document.write( "must be on the x-axis, because any point on the x-axis has a y-value of zero.
\n" ); document.write( ".
\n" ); document.write( "With this in mind we can return to the equation:
\n" ); document.write( ".
\n" ); document.write( "y = x(x+3)(x-1)
\n" ); document.write( ".
\n" ); document.write( "and see that there are three points on the x-axis. The first point is at x = 0, because
\n" ); document.write( "when x is equal to zero, y also equals zero. Therefore, (0, 0) is a point on the graph.
\n" ); document.write( ".
\n" ); document.write( "The second point is where x = -3 because when x = -3 the factor (x + 3) equals zero and
\n" ); document.write( "therefore y also equals zero. So the point (-3, 0) is on the graph.
\n" ); document.write( ".
\n" ); document.write( "The final point on the x-axis is where x = 1, because when x is equal to +1, the factor (x - 1)
\n" ); document.write( "is equal to zero and therefore y = 0. So the point (1, 0) is on the graph.
\n" ); document.write( ".
\n" ); document.write( "Now sketch a plot of these three points on a coordinate system ... on the x-axis put dots
\n" ); document.write( "at x = -3, at x = 0 (the origin), and at x = +1. The graph of the function must go through
\n" ); document.write( "all three of these points. Therefore, as your eyes move from left to right, the graph must
\n" ); document.write( "have one of two general shapes as noted below:
\n" ); document.write( ".
\n" ); document.write( "(1) The graph could come downward through the point at x = -3, then between x = -3 and
\n" ); document.write( "x = 0 it would drop a little then curve and turn upward to go through the point where x = 0,
\n" ); document.write( "following which it would rise a little then curve downward and go back downward through the
\n" ); document.write( "point where x = 1 and continue downward. It can never return to cross the x-axis again.
\n" ); document.write( "Or the graph could have another shape:
\n" ); document.write( ".
\n" ); document.write( "(2) The graph could come upward through the point at x = -3, then between x = -3 and
\n" ); document.write( "x = 0 it would rise a little then curve and turn downward to go through the point where x = 0,
\n" ); document.write( "following which it would sink a little then curve upward and go back up through the
\n" ); document.write( "point where x = 1 and continue upward. It can never return to cross the x-axis again.
\n" ); document.write( ".
\n" ); document.write( "We have these two possibilities. The following graph shows them:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "The green graph depicts the graph of case (1) above and the red graph depicts the graph
\n" ); document.write( "described in case (2).
\n" ); document.write( ".
\n" ); document.write( "It is fairly easy to find out which case applies. One way to do it is to pick a convenient value
\n" ); document.write( "for x between two of the points on the x-axis, substitute this value into the equation
\n" ); document.write( "and find out if y is positive or negative. For example, a convenient value of x might be
\n" ); document.write( "x = -1. It is between the x-axis crossings at x = -3 and x = 0. Substitute x = -1 into
\n" ); document.write( "the equation for y and you have:
\n" ); document.write( ".
\n" ); document.write( "y = x(x + 3)(x - 1) = -1*(-1 + 3)*(-1 -1) = -1*(+2)*(-2) = +4
\n" ); document.write( ".
\n" ); document.write( "This tells us that the point (-1, +4) is on the graph (y has a positive value when x = -1).
\n" ); document.write( ".
\n" ); document.write( "This being the case, the green graph is not the correct one. Therefore, the red graph applies
\n" ); document.write( "and we are working with the graph:
\n" ); document.write( ".
\n" ); document.write( "
\n" ); document.write( ".
\n" ); document.write( "The problem asked you to set up inequalities to show the values of x where f(x) (or its
\n" ); document.write( "equivalent y) is greater than zero.
\n" ); document.write( ".
\n" ); document.write( "Even without the exact graph we can now tell that f(x) or y is greater than zero when
\n" ); document.write( "x is between -3 and 0. So one answer is -3 < x < 0. And y is greater than zero when x
\n" ); document.write( "is greater than +1. So another answer is x > +1
\n" ); document.write( ".
\n" ); document.write( "Those are the two regions on the x-axis where y (or f(x)) is greater than zero.
\n" ); document.write( ".
\n" ); document.write( "I hope this helps you to understand the problem a little better and see how you can work it.
\n" ); document.write( ".
\n" ); document.write( " \n" ); document.write( "