Let's look at another problem related to finding roots of a quadratic equation.
Example 1. Investigate the signs of the roots of the quadratic equation
(a - 4)x2 - 2ax + a + 4 = 0
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 x2 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
To solve this inequality we use the intervalsmethod 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
of (a + 4)/(a - 4) on the obtained intervals.
To solve an inequality means to find its solution set , all the values of the
x which satisfy the inequality. Two inequalities are equivalent if they hold for
same values of x, that is, if their solution sets coincide. In the process of
an inequality we are trying to reduce it to a simpler equivalent inequality.
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.
f(x) > g(x) f(x) + a > g(x) + a
We can multiple both sides of an inequality by a positive number a.
if a > 0, f(x) > g(x) af(x) > ag(x)
•If we multiply both sides of an inequality by a negative
number a, the in-
if a < 0, f(x) > g(x) af(x) < ag(x)
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.
This method is used to solve inequalities of the form
Here the sign ">" could be replaced with "<",
" ≥ ", or " ≤ ".
(a) Find the points where f(x) or g(x) are either zero or change sign . If f(x)
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.
(b) 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 x2 - 5x + 6
here. Notice that if we do
this we have to consider cases when x2 - 5x + 6 > 0 and x2
- 5x + 6 < 0 so
we know if the inequality switches sign or not. This is doable, but let's
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
Answer: (-∞, ∞] ∪ (2, 3) ∪ [4,∞)
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
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: (-∞,-4) ∪ (0, 3/2]
Example 4. Solve the inequality
2x2 + 3x + 4 ≥ 0
This quadratic function has negative discriminant and hence has no x- intercepts .
The graph is a parabola that opens upward, hence the function is always
Example 5. Solve the inequality
3x2 - 4x + 2 < 0
This quadratic function has negative discriminant and hence has no
graph is a parabola that opens upward, hence the function is never negative.
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