Question 70474
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:
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{{{x^2 - 2x - 3 = 0}}}
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This is a typical quadratic equation.  This particular equation can be factored so that the
equation can be broken down to:
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{{{(x-3)*(x+1) = 0}}}
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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:
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First {{{x-3 = 0}}} and by adding 3 to both sides this becomes {{{x = 3}}}
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Then {{{x+1 = 0}}} and by subtracting 1 from both sides this becomes {{{x=-1}}}
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This tells us that for this problem you have critical points on the number line at {{{x=3}}}
and {{{x = -1}}}.   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.
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Return now to the original inequality equation as given in the problem statement:
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{{{x^2 - 2x - 3 > 0}}}
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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:
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{{{(-10)^2 - 2*(-10) - 3 > 0}}}
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{{{100 + 20 - 3 > 0}}}
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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.
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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:
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{{{(0)^2 - 2*(0) - 3 > 0}}} and this simplifies to: {{{-3 > 0}}}.
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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.)
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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.