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Solving Inequalities
Let's look at another problem related to roots of a
quadratic equation .
Example 1. Investigate the signs of the roots of the quadratic equation
depending on the value of the parameter a.
First of all we need to understand how many roots that equation has depending on
the
value of a. For this we find its discriminant
Hence the quadratic equation has two roots for every a. We
have to be careful here. When
a = 4 the coefficient of x ^{2} is zero and the equation is not quadratics anymore. In
this case
it has just one root x = 1 (check!) When a ≠ 4 using the quadratic formula we find
Hence one of the roots is always positive. Let's see when the second root is positive.
To solve this inequality we use the intervals method which
we will discuss in detail later.
We plot a = 4 and a = 4 on the number line with empty dots " o " (idicating that
these
points do not satisfy the ineqaulity) and figure out the signs of (a + 4)/(a  4)
on the
obtained intervals.
Hence the second root is positive if
. Similarly, the second root
is negative if a ∈ (4, 4).
We now know what is going on for each value of a except a = 4. In this case
the
product of roots is zero and hence one of the roots is zero. It is easy to see
that the other
root is 1.
Answer:
two positive roots
a ∈ (4, 4) one root is positive, another is negative
a = 4 unique root, positive
a = 4 one root is zero, another is negative
Inequalities occur in numerous other contexts. We now turn
to various techniques of
solving inequalities. Inequalities in one variable x are of the form
To solve an inequality means to find its solution set, all
the values of the variable x which
satisfy the inequality. Two inequalities are equivalent if they hold for the
same values of x,
that is, if their solution sets coincide. In the process of solving an
inequality we are trying
to reduce it to a simpler equivalent inequality. Recall that
means "equivalent".
Here are three basic rules for handling inequalities. The sign ">" below can be
replaced
with "<" ," ≥", or " ≤ ".
Rules for Handling Inequalities
•
We can add any constant a to both sides of an inequality.
• We can multiple both sides of an inequality by a positive number a.
if
• If we multiply both sides of an inequality by a negative
number a, the inequality
switches:
if
We will often use these rules to reduce our inequality to
the simplest form and will the
use the intervals method to solve the obtained inequality.
Intervals Method
This method is used to solve inequalities of the form
Here the sign ">" could be replaced with "<", "≥ ", or
"≤ ".
(1) Find the points where f(x) or g(x) are either zero or change sign . If f(x)
and
g(x) are polynomials, those points are the roots of f(x) = 0 and g(x) = 0. Mark
those points on the number line with a filled dot " •" if the point satisfies the
inequality and with an empty dot "o " otherwise.
(2) For each of the obtained intervals figure out the sign of f(x)/g(x) and write
that
sign next to the interval. Shade the values that satisfy the inequality.
Example 2. Solve the inequality
It is tempting to multiply both parts by x^{2}  5x + 6
here. Notice that if we do this we
have to consider cases when x^{2}  5x + 6 > 0 and x^{2}  5x + 6 < 0 so that we
know if the
inequality switches sign or not. This is doable, but let's
instead subtract 1/2 from both
sides and clear the denominators:
We are now ready to use the intervals method. Notice that
the roots of the numerator
x = 1 and x = 4 are marked with filled dots " • " as they satisfy the inequality.
Answer:
Notice how the signs alternate in the picture above. This is always the case if
both f(x)
and g(x) in the inequality f(x)/g(x) > 0 are polynomials that do not contain
factors of
the form (x  a)^{2n} (the degree is even). If either in f(x) or g(x) we have such
a factor
the sign of f(x)/g(x) does not change as we pass from one side of x = a to the
other.
Example 3. Solve the inequality
Here is the picture we get.
Notice that the sign does not change at x = 3 and x = 1 as
the corresponding terms
2x  6 and x  1 appear in the inequality in even degrees (4 and 2).
Answer:
Example 4. Solve the inequality
This quadratic function has negative discriminant and
hence has no xintercepts. The
graph is a parabola that opens upward, hence the function is always positive.
Answer:
Example 5. Solve the inequality
This quadratic function has negative discriminant and
hence has no xintercepts. The
graph is a parabola that opens upward, hence the function is never negative.
Answer: ,
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