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Chapter 2: Problem 12

Find the key numbers of the expression. $$\frac{x}{x+2}-\frac{2}{x-1}$$

Short Answer

Expert verified
The key numbers of the given expression are \(x = -2\) and \(x = 1\). These are the values that would make the denominator in the expression equal to zero.

Step by step solution

01

Identify the Denominators

This is a rational expression with two fractions. Each fraction has its own denominator. The denominators are \(x+2\) for the first fraction and \(x-1\) for the second fraction.
02

Set Each Denominator Equal to Zero

To find the key numbers, solve for x in each equation: 1. \(x+2 = 0\)2. \(x - 1 = 0\)
03

Solve for x in Each Equation

For Equation 1, subtract 2 from both sides to isolate x:\(x = -2\)For Equation 2, add 1 to both sides to isolate x:\(x = 1\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Denominators
In the world of fractions, denominators are equally as essential as numerators. A denominator is the bottom part of a fraction and it signifies the number of equal parts the whole is divided into. In rational expressions, like the one involved in our problem, the denominators are expressions in themselves.
To address the given exercise, each fraction has its own distinct denominator.
  • In the expression \( \frac{x}{x+2} \), the denominator is \( x+2 \).
  • For \( \frac{2}{x-1} \), the denominator is \( x-1 \).
Knowing the denominators is crucial because they determine the potential values that \( x \) cannot take 鈥� which we will explore further next. Understanding this helps ensure accurate solutions and maintains the integrity of the expression.
Zero of Denominator
Discovering when a denominator equals zero is fundamental in understanding rational expressions. A rational expression becomes undefined when its denominator is zero, as division by zero is not possible in mathematics. To find out when the expression is undefined, we need to calculate the zero of each denominator.
Let's break it down:
  • For the denominator \( x + 2 \), you solve the equation \( x + 2 = 0 \). Subtracting 2 from both sides gives \( x = -2 \). So, if \( x = -2 \), the first fraction becomes undefined.
  • For \( x - 1 \), the corresponding equation \( x - 1 = 0 \) is solved by adding 1 to both sides to find \( x = 1 \). This value makes the second fraction undefined.
Hence, the values \( x = -2 \) and \( x = 1 \) are key numbers as they cause the denominators 鈥� and therefore the whole expression 鈥� to become undefined.
Solving Equations
Solving equations is all about isolating the variable to find its value. In the context of finding key numbers in rational expressions, this involves setting the denominators equal to zero and solving for \( x \). This method finds the points where the expression is undefined. It involves simple algebraic manipulations:
  • Start with the equations established from the denominators, \( x+2 = 0 \) and \( x-1 = 0 \).
  • For \( x+2 = 0 \), subtract 2 from both sides to get \( x = -2 \). This tells us \( x \) cannot be \(-2\) without causing division by zero.
  • For \( x-1 = 0 \), add 1 to both sides to isolate \( x \), finding \( x = 1 \).
By solving these equations, one obtains key numbers that are crucial for understanding the behavior of rational expressions. This simple process ensures that we avoid undefined expressions in calculations and equations.

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Most popular questions from this chapter

Sketch the graph of each quadratic function and compare it with the graph of \(y=x^{2}\) (a) \(f(x)=-\frac{1}{2}(x-2)^{2}+1\) (b) \(g(x)=\left[\frac{1}{2}(x-1)\right]^{2}-3\) (c) \(h(x)=-\frac{1}{2}(x+2)^{2}-1\) (d) \(k(x)=[2(x+1)]^{2}+4\)

Find two positive real numbers whose product is a maximum. The sum of the first and twice the second is 24.

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=x^{2}-30 x+225$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=-5 x^{3}+x^{2}-x+5$$

Write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. Vertex: (-2,-2)\(;\) point: (-1,0)
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