1. Review Session #1

2. Outline Inequalities Exponents Absolute Value

3. Properties of Inequalities Property 1 If a < b, then a + c < b + c. Example: 5 < 12, so 5 – 7 < 12 – 7; that is, –2 < 5. If a, b, and c are any real numbers, then the following properties are applicable. Interpretation: For any inequality, we can add or subtract any number we like across the entire inequality and the resulting inequality will still be true. Exercise 1: Find the set of real numbers x that satisfy each of the following: a) b)

4. Properties of Inequalities Property 2 If a 0, then ac < bc. Example: –3 < 9, so –3(2) < 9(2); that is, –6 < 18. Interpretation: For any inequality, we can multiply across the entire inequality by any positive number we like and the resulting inequality will still be true. Exercise 2: Find the set of real numbers x that satisfy each of the following: a) b) and

5. Properties of Inequalities Property 3 If a bc. Example: –3 < 9, so –3(– 2) < 9(– 2); that is, 6 < –18. Interpretation: For any inequality, we can multiply across the entire inequality by any negative number we like and the resulting inequality will still be true as long as we switch the direction of the inequality. Exercise 3: Find the set of real numbers x that satisfy.

6. Absolute Value Example: | 5 | = 5 but | –3 | = 3. The absolute value of a number a is denoted by |a| and is defined as follows: Interpretation: If a is non-negative, then the absolute value does nothing to change the value of a. However, if a is negative, the absolute value makes it positive. Exercise 4: Evaluate the following: a) b)

7. Absolute Value Properties Property 1 |ab| = |a| |b| If a and b are any real numbers, then the following properties are applicable. Exercise 5: Evaluate the following: a) b) Property 2

8. Absolute Value and Inequalities If a is any real number and f (x) is any function of x, then the following statements are true. 1. is equivalent to 2. is equivalent to or Example: Saying is equivalent to saying ; that is, x lies in the interval. Example: Saying is equivalent to saying or ; that is, x lies in the interval.

9. Absolute Value and Inequalities Exercise 6: Find the set of real numbers x that satisfy each of the following: a) b) Exercise 7: Find the set of real numbers x that satisfy each of the following: a) b)

10. Exponents Example: If b is any real number and n is any positive integer, then the expression b n (read “b to the power n”) is defined to be Example: NOTE: b 0 = 1 for any b.

11. Exponents If is any positive integer, then the expression b 1/n is defined to be the number that, when raised to the nth power, is equal to b. That is, If such a number exists, it is called the nth root of b, and is sometimes written as NOTE: The only time when the number b 1/n does not exist is when b < 0 and n is even.

12. Properties of Exponents Property 1 Example: If a, b, m, and n are any real numbers, then the following properties are applicable. Exercise 8: Simplify/evaluate each of the following: a) b)

13. Properties of Exponents Property 2 Example: Exercise 9: Simplify/evaluate each of the following: a) b)

14. Properties of Exponents Property 3 Example: Exercise 10: Simplify/evaluate each of the following: a) b)

15. Properties of Exponents Property 4 Example: Exercise 11: Simplify/evaluate each of the following: a) b)

16. Properties of Exponents Property 5 Example: Exercise 12: Simplify/evaluate each of the following: a) b)