Ericka Clarice Wei
04/ SH1 Considerate
Math Summary
Irrational Inequality
1. Irrational Inequality - Definition
Irrational Inequality is an inequality that contains a variable under the sign of root n-degree. Any
equation where the variable is inside a radical is called an irrational inequality equation. The
removal of the radical will be the most important step in solving an irrational equation. To do so,
we would have to isolate each radical in one of the equation's sides, then square both sides.
Furthermore, we should always double-check the solutions because some of them may be
incorrect.
2. Finding sets of the solutions from irrational inequality equations
To be able to solve irrational inequality equations, you must have a perfect understanding of the
theory from several prior topics, particularly “Irrational equations” and “Rational inequalities.”
Let’s write down one of the most important theorems for dealing with irrational inequalities (that
is, inequalities with roots). As a result, if both f (x) and g (x) are non-negative, the inequality is:
f(x) > g(x) , equivalent to the following inequality : (f (x))” > (g (x))”
To put it another way, assuming the left and right inequalities are non-negative expressions, this
inequality can be safely raised to any degree. If you wish to boost all inequality to an unusual
degree, you don't even need the left and right sides of the inequality to be non-negativity in this
situation. As a result, any unrestricted inequality might be amplified to an unusual degree. We
underline once more that in order to achieve a level of equality, both sides of the inequality must
be non-negative.
This theorem is particularly useful in irrational inequalities, i.e. inequalities with roots, where it is
required to increase inequalities to some degree in order to solve most of the situations. Of
course, in the case of irrational inequalities, DHS must be carefully considered, as it is primarily
composed of two standard conditions:
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Even degree roots must contain non-negative expressions,
and fraction designations should not be zeros.
Also keep in mind that the root value of an even degree is never negative.
According to the preceding, if the irrational inequality contains more than two square roots, you
must ensure that there are non-negative expressions on each side of the inequality, i.e. sums of
square roots, before squaring the inequality in the square (or another even power). If there is a
difference in the roots on one side of the inequality, nothing can be known in advance about the
sign of the difference, and hence raising the inequality to an even degree is impossible.
The roots in front of which the minus signs are transferred to opposite sides of the inequality
(from left to right or vice versa) are required in this case; thus, the minus signs in front of the
roots will change to "pluses," and only sums of roots will be obtained from both sides of the
inequality. All inequalities can then be squared.
The method of shifting variables can be used to solve irrational inequalities, just as it can be
used to solve other issues in mathematics. The most important thing to remember is that after
a replacement, the new expression should become simpler and should no longer contain the old
variable. It's also important to remember to do a reverse replacement.
Let's look at a few simple yet common examples of irrational inequalities. When two roots of
even degree are compared, the first class of inequalities occurs, resulting in an inequality of the
form:
Because this inequality has non-negative expressions on both sides, it may be safely increased
to the power of 2 n, after which, when the LDL is taken into account, we get:
It's worth noting that the TLD is only written for that smaller expression. Because it is greater
than the first expression, which is greater than zero, another expression will inevitably be greater
than zero.
When the root of an even degree is supposed to be greater than a rational expression, i.e. when
there is an illogical inequality of the following form:
Moving to a combination of two systems is the solution to this inequality:
Finally, in the case when the root of an even degree is assumed to be less than some
rational expression , i.e. in the case when there is an irrational inequality of the form:
Moving to the system is the solution to this inequality:
When two roots of an odd degree are compared, or when the root of an odd degree is supposed
to be bigger or less than a rational expression, you can simply elevate all inequalities to the
desired odd degree, removing all roots. Since inequalities can be raised to an odd degree
without restriction and expressions of any sign can stand under the roots of odd degrees, there
is no additional DHS in this case.
3. Solving irrational inequality problems
Now, I will be solving these problems. I will attach the solution below.