Solving
Quadratic Inequalities: Examples (page
3 of 3)

Solvex^{2}
+ 2x – 8<0.

First, I'll find the
zeroes:

x^{2}
+ 2x – 8 = 0
(x
+ 4)(x – 2) = 0
x
= –4 or
x
= 2

These x-intercepts
split the number line into three intervals: x
< –4, –4 < x < 2,
and x
> 2. Since this
is a "less than" inequality, I need the intervals where the
parabola is below the x-axis.
Since the graph of y
= x^{2} + 2x – 8
is a right-side-up parabola, it is below the axis in the middle (between
the two intercepts). Since this is an "or equal to" inequality,
the boundary points of the intervals (the intercepts themselves) are
included in the solution.

Then the solution is:
–4<x<2

You can check the
answer from the graph:

There is one fiddly case
that you might not even have to deal with, but I'll cover it anyway, just
in case your teacher likes tricky test problems.

Solve–x^{2}
+ 6x – 9>0.

First, I'll find the
zeroes of y
= –x^{2} + 6x – 9,
the associated quadratic equation:

In this case, there
is exactly one x-intercept.
When you have only one intercept like this, the quadratic doesn't
cross the axis, but instead just touches it, as you can see here:

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I need to find where y
= –x^{2} + 6x – 9
is above the axis. But I know (and can verify from the above graph)
that this quadratic only touches the axis from below; it is never fully
above
the axis. However, this inequality is an "or equal to" inequality,
so the "equal" part counts as part of the solution. That is,
the intercept is part of the solution. In this case, it is actually
the only solution, because the graph only touches the axis (is equal
to zero); it never goes above (is never greater than zero).

So the solution is

x
= 3

By the way, if you're supposed
to write your solutions in set notation, a single-element set like this
is written as "{3}"
(those are curly-braces, not parentheses), and is called a "singleton
set".

In contrast to the previous
solution, look at this:

If you are careful about
finding the zeroes of the quadratic, and use your knowledge of the shape
of quadratic graphs, you shouldn't have any trouble solving quadratic
inequalities.