(2x-5)/(2-x)

4 min read Jun 16, 2024

Exploring the Rational Expression: (2x-5)/(2-x)

This article will delve into the rational expression (2x-5)/(2-x), exploring its key features, simplifying it, and analyzing its behavior.

Understanding the Expression

The expression (2x-5)/(2-x) is a rational expression. This means it is a fraction where both the numerator and denominator are polynomials.

Numerator: 2x - 5
Denominator: 2 - x

Simplifying the Expression

We can't simplify this expression directly by canceling terms. However, we can factor out a -1 from the denominator to get:

(2x-5)/(-1(x-2))

This allows us to rewrite the expression as:

-(2x-5)/(x-2)

Analyzing the Behavior

Now let's examine some key aspects of this expression:

Domain: The expression is undefined when the denominator is zero. Therefore, the domain is all real numbers except x = 2.
Vertical Asymptote: At x = 2, the denominator becomes zero, creating a vertical asymptote. The function approaches infinity as x approaches 2 from both sides.
Horizontal Asymptote: We can determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Both have a degree of 1. Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients, which is -2/1 = -2.
X-intercept: To find the x-intercept, we set the numerator equal to zero and solve for x:

2x - 5 = 0 x = 5/2

Therefore, the x-intercept is (5/2, 0).
Y-intercept: To find the y-intercept, we set x = 0 and evaluate the expression:

-(2(0)-5)/(0-2) = 5/2

Therefore, the y-intercept is (0, 5/2).

Graphical Representation

By plotting the intercepts, asymptotes, and a few additional points, we can sketch the graph of the function. This will show us the behavior of the expression for different values of x.

Applications

Rational expressions like this are commonly used in various areas of mathematics and science, including:

Calculus: To find limits, derivatives, and integrals.
Physics: To model physical phenomena like motion, electricity, and magnetism.
Engineering: To design and analyze structures, circuits, and systems.

Understanding the characteristics and behavior of rational expressions is essential for solving problems in these fields.