# 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 x-intercepts. 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 x-intercepts. The

graph is a parabola that opens upward, hence the function is never negative.

Answer: ,

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