You can
put this solution on YOUR website!Temporarily replace the inequality sign with an equal sign ... just because you are probably
more familiar with working on equations. This temporary maneuver results in:
.
.
This is a typical quadratic equation. This particular equation can be factored so that the
equation can be broken down to:
.
.
This equation will be true if either of the factors on the left side is zero, because that
would make the left side of the equation equal 0 ... the same value that is on the right side.
So, one at a time set the two factors equal to zero and solve for x:
.
First
and by adding 3 to both sides this becomes
.
Then
and by subtracting 1 from both sides this becomes
.
This tells us that for this problem you have critical points on the number line at
and
. Draw a horizontal number line and mark the points +3 and -1 on that line.
There are three important regions on this number line ... the region extending to the left
from the point -1, the region between the points -1 and +3, and the region extending to
the right from the point x = +3. You need to examine just these three regions to tell which of
them satisfy the inequality equation.
.
Return now to the original inequality equation as given in the problem statement:
.
.
Now pick any convenient number that is in the first region to the left of x=-1. Suppose that
you choose x = -10. Plug that value in for x in the inequality equation and simplify the
results. The substitution of -10 leads to:
.
.
.
You can already see that this equation is true because 117 > 0. Now you know that numbers
in the region from -1 all the way down to negative infinity (but not including -1) will satisfy
the inequality equation.
.
Next examine the region with -1 on one end and +3 on the other end. Select a convenient value
of x between these two end points and see what happens to the inequality equation. A
convenient value is x = 0 because when it is substituted in for x, it cause terms that
have an x in them to become zero. The substitution is:
.
and this simplifies to:
.
.
This is obviously not true. So all the points from -1 to +3 will not make the inequality
equation true. Therefore, x cannot be from -1 to +3 including the end points. (Plug -1
and +3 into the inequality equation and see if the result is GREATER than zero.)
.
Finally do the same exercise for a single convenient value in the region where x is greater
than +3. It might be convenient to use x = +10, but you can use any value greater
than +3. If you do this you will find that it does satisfy the inequality equation,
so all values in that region of x greater than +3 all the way out to + infinity will satisfy
the equation.
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In summary the inequality equation is satisfied if x < -1 or x > +3.
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Hopefully this refreshes your memory on problems such as these.
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