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A Recap/Introduction to Functions, Equations, Inequalities, and Notation

A Recap/Introduction to Functions, Equations, Inequalities, and Notation. Properties of Equality. 1. If. =. , then. +. =. +. Addition Property. a. b. a. c. b. c. 2. If. =. , then. –. =. –. Subtraction Property. a. b. a. c. b. c. 3. If. =. , then. =.

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A Recap/Introduction to Functions, Equations, Inequalities, and Notation

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  1. A Recap/Introduction to Functions, Equations, Inequalities, and Notation

  2. Properties of Equality 1. If = , then + = + . Addition Property a b a c b c 2. If = , then – = – . Subtraction Property a b a c b c 3. If = , then = ,  0. Multiplication Property a b ca cb c a b 4. If = , then = ,  0. Division Property a b c c c 5. I f = , then either may replace the other in Substitution Property a b any statement without changing the truth or falsity of the statement. 1-1-1

  3. Strategy for Solving Word Problems 1. Read the problem carefully—several times if necessary; that is, until you understand the problem, know what is to be found, and know what is given. 2.Let one of the unknown quantities be represented by a variable, say x, and try to represent all other unknown quantities in terms of x. This is an important step and must be done carefully. 3. If appropriate, draw figures or diagrams and label known and unknown parts. 4.Look for formulas connecting the known quantities with the unknown quantities. 5. Form an equation relating the unknown quantities to the known quantities. 6.Solve the equation and write answers to all questions asked in the problem. 7. Check and interpret all solutions in terms of the original problem—not just the equation found in step 5—since a mistake may have been made in setting up the equation in step 5. 1-1-2

  4. Quantity-Rate-Time Formulas 1-1-3

  5. Systems of Linear Equations 1-2-4

  6. [ ] x a b [ ) x a b ( ] x a b ( ) x a b Interval Notation Interval Inequality Notation Notation Line Graph Type [ a, b ] a  x  b Closed [ a, b ) a  x < b Half-open ( a, b ] a < x  b Half-open ( a, b ) a < x < b Open 1-3-5-1

  7. Interval Notation Interval Inequality Notation Notation Line Graph Type x [  [ b , ) x  b Closed b x ( b,  ( ) x > b Open b x ] –, ( a ] x  a Closed a ) x , ( – a ) x < a Open a 1-3-5-2

  8. Inequality Properties For a, b, and c any real numbers: 1. If < and < , then < . Transitive Property a b b c a c 2. If < , then + < + . Addition Property a b a c b c 3. If < , then – < – . Subtraction Property a b a c b c 4. If < and is positive, then < . a b c ca cb Multiplication Property ü ý (Note difference between þ 4 and 5. ) 5. If < and is negative, then > . a b c ca cb a b ü Division Property 6. If < and is positive, then < . a b c c c ý (Note difference between þ 6 and 7. ) a b 7. If < and is negative, then > . a b c c c 1-3-6

  9. 0 < | – c | < Absolute Value Equations and Inequalities | x – c | = d { c – d , c + d } | x – c | < d ( c – d , c + d ) ( – d c )  ( c , + ) x d c , c d (   , c – )  ( c d  ) | x – c | > d d + , 1-4-7

  10. Particular Kinds of Complex Numbers Imaginary Unit: i and real numbers Complex Number: + a bi a b Imaginary Number: +  0 a bi b Pure Imaginary Number: 0 + =  0 bi bi b Real Number: + 0 = a i a Zero: 0 + 0 = 0 i Conjugate of – a bi : a bi + 1-5-8

  11. Subsets of the Set of Complex Numbers Natural numbers (N) Integers (Z) Zero Rational numbers (Q) Negative Real Noninteger Integers numbers (R) rational numbers Irrational Complex numbers (I) numbers (C) Imaginary numbers Ì Ì Ì Ì N Z Q R C 1-5-9

  12. Discrimant Roots of + + = 0 ax bx Quadratic Formula Discriminant and Roots 2 c 2 – 4 , , and c real numbers, a  0 b ac a b 1-6-10

  13. Squaring Operation on Equations Equation Solution Set x= 3 {3}x2= 9 {–3, 3} 1-7-11

  14. Key Steps in Solving Polynomial Inequalities Step 1. Write the polynomial inequality in standard form (a form where the right-hand side is 0.) Step 2. Find all real zeros of the polynomial (the left side of the standard form.) Step 3. Plot the real zeros on a number line, dividing the number line into intervals. Step 4. Choose a test number (that is easy to compute with) in each interval, and evaluate the polynomial for each number (a small table is useful.) Step 5. Use the results of step 4 to construct a sign chart, showing the sign of the polynomial in each interval. Step 6. From the sign chart, write down the solution of the original polynomial inequality (and draw the graph, if required.) 1-8-12

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