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When solving inequalities, expressing solutions using interval notation is a common method. Interval notation helps in understanding a range of solutions without listing every possible value. For example, if a solution includes all numbers from -1 to 3, it would be expressed as

• \((-1, 3)\)

This indicates that all numbers between, but not including, -1 and 3, are solutions to the inequality. The use of round brackets

• \(()\)

signifies that the endpoints are not part of the solution set, often due to a zero in the denominator making the expression undefined. In contrast, square brackets

• \([ ]\)

include the endpoints in the solution set, usually when the inequality includes an equal sign. For inequalities where a variable could take positive infinity (∞) or negative infinity (-∞), these are always written with parentheses, since infinity is not a specific number that can be reached. In the original solution, the interval notation

• \((-\infty, 1) \cup (1, +\infty)\)

was used to indicate that the solution includes all real numbers except 1, signaling a discontinuity or undefined point at \(x = 1\). This method provides a precise and clear way to convey a range of possible solutions.

Rational functions are expressions of the form

• \(\frac{P(x)}{Q(x)}\)

where both \(P(x)\) and \(Q(x)\) are polynomials, with \(Q(x) eq 0\). They are called 'rational' because they relate closely to ratios and have a structure of fraction with a polynomial numerator and denominator. Solving inequalities involving rational functions, like the one in this exercise, frequently requires us to determine where the function's output is positive or negative. The function given in the exercise is

• \(\frac{5}{(1-x)^2}\)

This is a simple rational function with a constant numerator and a polynomial denominator. The key to solving inequalities involving rational functions is finding critical points where the denominator is zero, as these points might cause the function to be undefined or change signs. Once these critical values are determined, the real number line can be divided into intervals. Each interval can then be tested to determine if the function is positive or negative in that region. Since

• \((1-x)^2 \ge 0\)

for all real \(x\), and \(\frac{5}{(1-x)^2}\) becomes undefined at \(x = 1\), identifying and excluding such points aids in correctly defining the solution set.

The concept of real numbers is foundational in mathematics. Real numbers include all the numbers on the number line, encompassing both rational and irrational numbers.

• Rational numbers are those that can be expressed as a fraction of two integers, like \(\frac{3}{4}\) or \(-2\).
• Irrational numbers are those that cannot be expressed as such fractions, such as \(\pi\) or \(\sqrt{2}\).

In the context of solving inequalities, real numbers represent all possible solutions unless a specific number is excluded due to causing an undefined condition.In this exercise, the inequality

• \(\frac{5}{(1-x)^{2}} > 0\)

is satisfied by all real numbers except 1. This is because at \(x = 1\), the denominator becomes zero, making the function undefined at that particular point. Understanding real numbers and their properties enables us to effectively interpret the solutions of inequalities and rational functions. This also involves excluding any points from the solution set where the function ceases to be real or becomes undefined. Every other number on the real number line is valid as a potential solution to the inequality, provided that it does not make the denominator zero.