Rational Expressions and Equations Chapter 8 Rational Expressions and Equations

Chapter Sections 8.1 – Simplifying Rational Expressions 8.2 – Multiplication and Division of Rational Expressions 8.3 – Addition and Subtraction of Rational Expressions with a Common Denominator 8.4 – Addition and Subtraction of Rational Expressions 8.5 – Complex Fractions 8.6 – Solving Rational Equations 8.7 – Rational Equations: Applications & Problem Solving 8.8 – Variation Chapter 1 Outline

Simplifying Rational Expressions § 8.1 Simplifying Rational Expressions

Rational Expressions A rational expression is an expression of the form p/q where p and q are polynomials and q  0. Examples: Whenever a rational expression contains a variable in the denominator, assume that the values that make the denominator 0 are excluded.

Signs of a Fraction - a + + b Three signs are associated with any fraction: the sign of the numerator, the sign of the denominator, and the sign of the fraction. sign of the numerator - a + b + sign of the fraction sign of the denominator Changing any two of the three signs of a fraction does not change the value of the fraction - a b a -b = a b - =

Simplifying A rational expression is simplified or reduced to lowest terms when the numerator and denominator have no common factors other than 1. Examples: a) b)

Simplifying Rational Expressions Factor both the numerator and denominator as completely as possible. Divide out any factors common to both the numerator and denominator. Simplify

Factoring a Negative 1 Remember that when –1 is factored from a polynomial, the sign of each term in the polynomial changes. Example: – 2x + 5 = – 1(2x – 5) = –(2x – 5) Simplify

Multiplication and Division of Rational Expressions § 8.2 Multiplication and Division of Rational Expressions

Multiplying Fractions - 5 21 · 3 4 1 - 5 21 · 3 4 = - 5 21 · 3 4 - 5 7 · 1 4 = - 5 28 7

Multiplying Rational Expressions Factor all numerators and denominators completely. Divide out common factors. Multiply numerators together and multiply denominators together. Multiply

Dividing Two Fractions Divide . - 2 9  5 1 = - 2 9  5 - 2 9 · 5 = - 2 9 · 5 - 2 5 = 1

Dividing Rational Expressions Invert the divisor (the second fraction) and multiply Divide

§ 8.3 Addition and Subtraction of Rational Expressions with a Common Denominator

Adding/Subtracting Fractions 5 12 2 + 7 12 = 5 2 +

Common Denominators Subtract Add or subtract the numerators. Place the sum or difference of the numerators found in step 1 over the common denominator. Simplify the fraction if possible. Subtract

Common Denominators Example: a.) Add

Common Denominators Example: b.) Subtract

Least Common Denominator Factor each denominator completely. Any factors used more than once should be expressed as powers. List all different factors that appear in any of the denominators. When the same factor appears in more than one denominator, write that factor with the highest power that appears. The least common denominator (LCD) is the product of all the factors listed in step 2.

Least Common Denominator Find the LCD: a.) The LCD is (2x + 5)(x – 5). b.) The LCD is x(x + 1). c.) The LCD is 36w5z4.

Addition and Subtraction of Rational Expressions § 8.4 Addition and Subtraction of Rational Expressions

Unlike Denominators Determine the LCD. Rewrite each fraction as an equivalent fraction with the LCD. Add or subtract the numerators while maintaining the LCD. When possible, factor the remaining numerator and simplify the fraction.

Unlike Denominators Example: a.) The LCD is w(w+2).

This cannot be factored any further. Unlike Denominators Example: b.) The LCD is 12x(x – 1). This cannot be factored any further.

Unlike Denominators Example: c.) The LCD is (x – 5)(x+2).

§ 8.5 Complex Fractions

Simplifying Complex Fractions A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator. Example:

Simplify by Combining Like Terms Method 1 Add or subtract the fraction in both the numerator and denominator of the complex fraction to obtain single fractions in both the numerator and the denominator. Invert the denominator of the complex fraction and multiply the numerator by it. Simplify further if possible.

Simplify by Combining Like Terms a) b)

Simplify by Multiplying Method 2 Find the LCD of all the denominators appearing in the complex fraction. Multiply both the numerator and the denominator of the complex fraction by the LCD. Simplify further if possible.

Simplify by Multiplying a) LCD is 4b3. b) y(x-y) LCD is y(x-y).

Solving Rational Equations § 8.6 Solving Rational Equations

Complex Fractions A rational equation is one that contains one or more rational (fractional) expressions. Example:

Solving Rational Equations Find the LCD of all fractions in the equation. Multiply both sides of the equation by the LCD. (Every term will be multiplied by the LCD.) Remove any parentheses and combine like terms on each side of the equation. Solve the equation. Check the solution in the original equation.

Integer Denominators Solve the equation: a) The LCD is 30.  CHECK:

Variable Denominators Whenever there is a variable in the denominator, it is necessary to check your answer in the original equation. If the answer obtained makes the denominator zero, that value is NOT a solution to the equation. Solve: The LCD is 2z. CHECK: 

Variable Denominators Solve: The LCD is 2(x-3). No solution CHECK:

Rational Equations: Applications & Problem Solving § 8.7 Rational Equations: Applications & Problem Solving

Geometry Applications Yield signs are triangles. The area of the sign is about 558 square inches. The height of the sign is about 5 inches less than its base. Determine the length of the base of the yield sign. h = b – 5 b A=bh Solve the equation.

Geometry Applications The base is 36 inches. CHECK: 

Work Problems Problems in which two or more people or machines work together to complete a certain task are referred to as work problems. part of task done by first machine part of task done by second machine 1 (one whole task completed) + =

Work Problems Example: At the NCNB Savings Bank it takes a computer 4 hours to process and print payroll checks. When a second computer is used and the two computers work together, the checks can be printed in 3 hours. How long would it take the second computer by itself to process and print the payroll checks? Continued.

A table helps to keep information organized. Work Problems Example continued: Machine Rate of Work Time Part of Task (rate x time) 1st 2nd A table helps to keep information organized. Continued.

Work Problems Example continued: Machine Rate of Work Time Part of Task (rate x time) 1st 2nd One job done Solve the equation. Continued.

Work Problems Example continued:  The LCD is 4x. It would take the second computer 12 hours by itself. CHECK: 

§ 8.8 Variation

Variation A variation is an equation that relates one variable to one or more other variables using the operations of multiplication or division. There are two different types of variation: direct variation and inverse variation

Direct Variation The formula used to calculate distance is distance = rate  time If the rate is a constant 50 miles per hour, the formula can be written as d = 50t The distance, d, varies directly as the time, t, so the distance is directly proportional to the time, t. The 50 is the constant of proportionality.

Direct Variation If a variable y varies directly as a variable x, then y = kx where k is the constant of proportionality (or the variation constant).

Direct Variation Example: s varies directly as the square of m. If s = 100 when m = 5, find s when m = 12. First, determine what k represents. s = km2 100 = k(52) 100 = 25k 4 = k Then determine what s is when m = 12. s = km2 s = 4(122) s = 576

Indirect Variation If a variable y varies inversely as a variable x, then where k is the constant of proportionality. k y = or xy = k x Example: R varies inversely as W. Find R when W = 160 and k = 240.