MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE. A rational expression is just a fraction involving polynomials, for example 3x2 2

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1 MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE SECTION 1.2: PRECALCULUS REVIEW II Practice: 3, 7, 13, 17, 19, 23, 29, 33, 43, 45, 51, 57, 69, 81, Rational Expressions and Other Algebraic Fractions A rational expression is just a fraction involving polynomials, for example 3x2 2 is a rational x + 7 expression. Like regular fractions, rational expressions have many different forms, and we generally want them written in lowest terms. That is, the expression should be simplified so that all identical factors on both top and bottom have been cancelled out. Note that if you have a fraction where the expressions are not polynomials, we call this an algebraic fraction and we use the same techniques to simplify it. Example 1. Simplify the following: x2 3x + 2(2x 1) Answer: x2 3x + 2(2x 1) x2 3x + 4x 2 x2 + x 2 (x + 2)(x 1) x 2 + 4x 5 (x + 5)(x 1) x + 2 x + 5 Sometimes you have to combine fractions in order to simplify an expression, and you need to get common denominators. Example 2. Perform the indicated operation and simplify: x x 1 + 2x + 3 x 2 1 x Answer: x 1 + 2x + 3 x 2 1 x(x + 1) (x 1)(x + 1) + 2x + 3 (x + 1)(x 1) x2 x + 2x + 3 x2 + x + 3 (x + 1)(x 1) x 2 1 The top is not factorable, so we leave it in that form. 1

2 2 SECTION 1.2: PRECALCULUS REVIEW II Exercise 3. Perform the indicated operation and simplify: (a) x2 6x + 9 x 2 x 6 3x + 6 2x 2 7x + 3 reciprocal) (Note: division of a fraction is the same as multiplying by the (b) 1 x + 1 y (Hint: Combine the fractions on top and on bottom first by getting common denomi- 1 1 xy nators. Then multiply by the reciprocal and simplify). 2. Rationalizing algebraic fractions When the denominator of a fraction involves a difference or sum that has a square root in it, we have a technique to get rid of the square root from the denominator. This is called rationalizing, and the idea is that you multiply the top and bottom of the fraction by that same term, except you switch the plus to a minus (or vice versa). For example, if we had the term 5, and we multiplied by + 5, we get ( 5)( + 5) x x + 25 which has no square roots in it. Note that you can also rationalize a numerator. Example 4. Rationalize the numerator: x Answer: 3 x + 3 x (x + 3) x 3( ) 3 3( )

3 MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE 3 Exercise 5. Rationalize the denominator of Inequalities We solve inequalities in much the same way that we solve equations, with some minor but important differences. First of all, what you do to one side of an inequality you must do to the other, however if you multiply both sides by a negative number then you must reverse the inequality. Two examples of dealing with inequalities involve solving compound inequalities and inequalities involving zero. Example 6. Find all values of x that satisfy the inequality 0 < 3x Answer: Subtracting 7 from each entry, we get 7 < 3x 6. Then we divide through by 3 to get our final answer of 7/3 < x 2. Another way of writing the answer is to say all values of x in the interval ( 7/3, 2]. Example 7. Find all values of x that satisfy the inequality (2x 4)(x + 2) 0. Answer: When you are multiplying numbers, your answer will be greater than or equal to zero only when the two factors have the same sign. That is, when both numbers are positive or when both are negative. This means that for the inequality above to be true, we need both 2x 4 0 and x or 2x 4 0 and x In the first situation we solve each inequality to find that x 2 and x 2, and the only time when both of those inequalities is true is when x 2. In the second situation, we have x 2 and x 2, and this will only happen when x 2. So our final answer is when x 2 or x 2. Exercise 8. Find all values of x that satisfy the inequalities: (a) 2x 1 x + 2 4

4 4 SECTION 1.2: PRECALCULUS REVIEW II (b) 3x 4 0 (Note: be careful about your inequalities- you don t want to allow division by zero!) 2x Absolute Value Definition 9. The absolute value of a number n is denoted by n and is defined by n if n 0 n n if n < 0 Since the negative of a negative is positive, the absolute value of a number is always positive. It is important to know the definition above, it will show up later in the course many times. The most important thing to know is how to use the definition above to solve equations or inequalities involving absolute values. Example 10. Solve the inequality 5x Answer: For the inequality to be true, we know that (5x + 2) will either need to be between 0 and 3 (so taking the absolute value won t change anything and it will be less than or equal to 3) or (5x + 2) must be between -3 and 0 (then the absolute value will make it positive, but it will still be less than or equal to 3). Mathematically, we write this as 3 5x + 2 3, which solves down to 1 x 1/5 and is our answer. Exercise 11. Solve the inequality 2x

5 MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE 5 5. Applied Problems One of the focuses of this course is how calculus applies to real-world situations, so we will often have word problems to solve. The hardest part is reducing the words down to the correct math problem to solve. Once you have that, actually solving the problem is fairly straightforward (though not always). Exercise 12. The concentration (in mg/cc) of a certain drug in the bloodstream t hours after injection is given by 0.2t t Find the interval of time when the concentration of the drug is greater than or equal to 0.08 mg/cc. Hint: The words greater than or equal to tell you what you should be solving. Namely, you want the concentration to be 0.08, so you need to solve the inequality 0.2t t You may + 1 need the quadratic formula to find your answer.

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology Intermediate Algebra Gregg Waterman Oregon Institute of Technology c 207 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license