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Algebra: Linear Equations

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Section 6.1: Rational Expression
Which numbers cannot be used in place of the variable in each rational expression?
18. 2y+1
y² - y- 6

Section 6.2: Multiplication and Division
Perform the indicated operation
24. 3 * 2a+ 2
a² + a 6

Section 6.3: Finding the Least Common Denominator
Find the LCD for the given rational expressions, and convert each rational expression into an equivalent rational expression with the LCD as the denominator.
20. 4 * 5x
x-y 2y - 2x

Section 6.4: Addition and Subtraction of rational numbers
Perform the indicated operation. Reduce each answer to lowest terms.
8. 4 - 1
9 9

Section 6.5: Complex Fractions
Simplify each complex fraction. Reduce each answer to lowest terms.
52. 1 + 1
9 3x
X - 1
9 x

Section 6.6: Solving Equations with Rational Expressions
Solve each equation
30. x = 4
3 x + 1

Discussion Questions
DQ 1: I recently read an obituary of Paul MacCready, the father of human powered aircraft. He invented the Gossamer Albatross, an aircraft which successfully flew the English Channel under human power (a cyclist turning the propeller). The aircraft flew from Folkestone, England to Cap Griz Nez, France, a distance of 22 miles, in approximately 3 hours fighting a headwind. On the return trip, this headwind would become a tailwind. Let us assume that the cyclist can pedal at a rate of 12 mi / hr without wind interference. Exhibit a rational expression for the round trip time, Folkestone to Cap Griz Nez and back, as a function of wind speed. What would be the domain of this rational expression?

So, given a rational expression, the first issue which we may consider is to determine the domain of the expression. For instance, x-1 = 1 / x constitutes a rational expression, which has a domain consisting of all x ≠ 0. What would be the domain of the rational expression x = x / 1? Ans: the domain in this case is all values of x ( the real line ), as the denominator is never 0. For the case of a polynomial denominator in our rational expression, we note that the domain of the expression would be all values of x which do not cause our polynomial denominator to be zero. As an example, for our rational expression
1 / (2x + 1) we note that the zero of our linear (first degree) expression (2x + 1) is x = -1/2. Thus, the domain of this rational expression constitutes all values of x satisfying x ≠ -1/2.

DQ 2: What is the domain of the rational expression 1 / (x2 + 2x - 15 ) ?

DQ 3: What is the domain of the rational expression 1 / (x2 + 1), where x is considered to be a real number (imaginary or complex numbers not included in the domain) ?

As in the case of fractions, the simplification occurs by determining all prime factors of the numerator and denominator and deleting those prime factors which are common to both the numerator and denominator. As an example, we wish to simplify the rational expression (x2 + 2x + 1 ) / (x + 1).
Knowing that the numerator can be expressed as a perfect square, our rational expression becomes
(x + 1)2 / (x + 1) which, of course. reduces to the binomial x+1.
Of course, there is the issue that our original rational expression has as its domain all values of x for which x ≠ -1 and our final simplified rational expression is valid for all x. Technically, we would specify that the domain for our simplified expression would be the same as the domain of our original expression, that is all values of x for which x ≠ -1. Note that, if x = -1, then the original expression would be equivalent to the expression 0/0, which is undefined. An issue to remember is that, as in the case of integer fractions, only factors can be canceled from the numerator and denominator of a rational expression, so that factoring of our polynomial expressions becomes key to simplifying a rational expression.
DQ 4: Simplify the rational expression (2x2 + 13x + 20 ) / (2x2 + 17x + 30 ) . And, determine the domain of the rational expression.


Part I

Given the following two rational expressions

1 / (x2 - 1 ) , 1 / (x - 1)2

i determine the least common denominator (LCD) of the two expressions
ii expand each rational expression to its equivalent form having the LCD as its denominator
iii Add the two resulting rational expressions, simplify if possible, and determine the domain of the resulting rational expression

Part II

For the quadratic equation

x2 + x - 1 = 0.

i Determine the roots of the equation
ii Verify each root by plugging the root back into the equation and performing the computation

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Solution Summary

Operations with fractions, finding LCD