College Algebra Chapter 1 Equations and Inequalities College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Solving equations using Addition and multiplication properties of equality College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Solving equations using Addition and multiplication properties of equality General Approach I. Simplify the equation •Clear fractions or decimals as needed/desired •Combine any like terms II. Solve the equation •Use additive property of equality to write the equation with all variable terms on one side and constants on the other •Simplify each side •Use the multiplicative property of equality to obtain solution form College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Solving equations using Addition and multiplication properties of equality General Approach I. Simplify the equation •Clear fractions or decimals as needed/desired •Combine any like terms II. Solve the equation •Use additive property of equality to write the equation with all variable terms on one side and constants on the other •Simplify each side •Use the multiplicative property of equality to obtain solution form College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Solving equations using Addition and multiplication properties of equality General Approach I. Simplify the equation •Clear fractions or decimals as needed/desired •Combine any like terms II. Solve the equation •Use additive property of equality to write the equation with all variable terms on one side and constants on the other •Simplify each side •Use the multiplicative property of equality to obtain solution form College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Linear Equation Terminology Algebraic expression Sum or difference of algebraic terms Equation Statement that two expressions are equal Linear equations Three tests – exponents, divisors, multiplication Standard form Ax+By=C, where A and B are not both zero Conditional equation Equation is true for one value and false for all others Identity Equation that is always true Contradictions Equations that are never true College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving No Solution 0 College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving 5 1 x 3 x 9 3 x 1 9 3 33 x 25 1 1 x 7 10 x 9 3 3 0 50 College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Linear Equation Terminology Literal equation An equation that has two or more unknowns Formula Equation that models a known relationship between two or more quantities College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Literal Equations: V l w h solve for w a 2 b2 c 2 solve for b College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Literal Equations: d x2 x1 2 y2 y1 solve for x2 College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Consecutive Integer Problems Sum of 3 even integers is 66. What are they? College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Mixture Problems College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Mixture Problems College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Homework pg 80 1-86 College Algebra Chapter 1.2 Linear Inequalities in One Variable Inequalities and Solution Sets Terminology Solution Set The set of numbers that satisfy an inequality Set Notation Method of writing a solution set. Ex {x|x>k} Number Line Method of writing a solution set Interval Notation Method of writing a solution set. Ex Boundary Point x k , Marks the location of a number on a number line also called an endpoint Inclusion Uses brackets [ ] to indicate Not Included Uses parenthesis ( ) to indicate College Algebra Chapter 1.2 Linear Inequalities in One Variable Inequalities and Solution Sets Terminology Set Notation n | n 1 The set of all n Such that n is greater than or equal to 1 College Algebra Chapter 1.2 Linear Inequalities in One Variable Inequalities and Solution Sets Terminology Interval Notation n 1, n is an element of This set Smaller number to left Larger number to right [,] brackets if boundary points is included (,) parenthesis if boundary point is not included College Algebra Chapter 1.2 Linear Inequalities in One Variable Write the following in set notation, number line graph, and interval notation x is less than 10 Set notation x | x 10 Number line graph 10 Interval notation n ,10 College Algebra Chapter 1.2 Linear Inequalities in One Variable Write the following in set notation, number line graph, and interval notation n is greater than or equal to 2 Set notation n | n 2 Number line graph 2 Interval notation n 2, College Algebra Chapter 1.2 Linear Inequalities in One Variable Write the following in set notation, number line graph, and interval notation X is less than 5 and greater than or equal to -1 Set notation x | 1 x 5 Number line graph -1 Interval notation x 1,5 5 College Algebra Chapter 1.2 Linear Inequalities in One Variable Solving Inequalities Beware of multiplying or dividing by a (-) 3 p 2 12 8 80 p 3 2 1 5 x 3 2 6 x 1 2 College Algebra Chapter 1.2 Linear Inequalities in One Variable Solving Inequalities x 2 n 1 College Algebra Chapter 1.2 Linear Inequalities in One Variable Solving Compound Inequalities 2 x 0 10 k 8 College Algebra Chapter 1.2 Linear Inequalities in One Variable Solving Compound Inequalities r 8 or r 5 x 0 or x 8 College Algebra Chapter 1.2 Linear Inequalities in One Variable Homework pg 90 1-94 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Zero factor property Given that A and B represent real numbers or real-valued expressions, if A B 0 , then either A 0 or B 0 College Algebra Chapter 1.3 Solving Polynomial and Other Equations 2 x 20x 3x Solve 3 2 2 x 3x 20x 0 3 Set equal to zero 2 x 2 x 2 3x 20x 0 x2 x 5x 4 0 x 0 or 2 x 5 0 or x 4 0 5 x 0 or x or x 4 2 Remove common factors aka undistribute Factor trinomial Solve each factor equal to zero College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solve t 4t 7 54 t 13 or 2 College Algebra Chapter 1.3 Solving Polynomial and Other Equations x 2 15 2 x x5 x 3 a 2 17 8 a3 a 3 m2 8m 16 m4 b 2 5b 24 b 8 b3 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 1. Identify and exclude values that cause a zero denominator 2. Multiply by the LCD and simplify 3. Solve resulting equation 4. Check in original equation and step 1 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 1. 2 1 4 2 m m 1 m m m6 2. 3. 4. Identify and exclude values that cause a zero denominator Multiply by the LCD and simplify Solve resulting equation Check in original equation and step 1 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 1. 14 2x x 1 x7 x7 x 3 or x 7 x7 2. 3. 4. Identify and exclude values that cause a zero denominator Multiply by the LCD and simplify Solve resulting equation Check in original equation and step 1 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 1. 2. 3. 4. Identify and exclude values that cause a zero denominator Multiply by the LCD and simplify Solve resulting equation Check in original equation and step 1 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 1. 2. 3. 4. Identify and exclude values that cause a zero denominator Multiply by the LCD and simplify Solve resulting equation Check in original equation and step 1 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations 1. Isolate radical term 2. Square both sides 3. Simplify College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations x 1 12 10 3 1. 2. 3. Isolate radical term Square both sides Simplify College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations 1. 2. 3. Isolate radical term Square both sides Simplify College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations 1. 2. 3. Isolate radical term Square both sides Simplify College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations x 2 2x 4 10 1. 2. 3. Isolate radical term Square both sides Simplify College Algebra Chapter 1.3 Solving Polynomial and Other Equations Rational Exponents 3x 1 9 15 3 4 15 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Rational Exponents 8x 3 2 11 17 8 16 x 25 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving with U-Substitution 2 3 1 3 x 3x 10 0 x 125 or x 8 College Algebra Chapter 1.3 Solving Polynomial and Other Equations Homework pg 101 1-100 College Algebra Chapter 1.4 Complex Numbers Imaginary unit i represents the number whose square is -1 i 2 1 i 1 College Algebra Chapter 1.4 Complex Numbers i 1 i 1 2 i 3 1 1 i4 1 i 5 i 4 i 1 College Algebra Chapter 1.4 Complex Numbers Complex numbers are those that can be written in the form a bi , where a and b are real numbers and i 1 . The expression a bi is called the standard form of a complex numbers a bi Real part Imaginary part College Algebra Chapter 1.4 Complex Numbers Identify the values of a and b a=-5 -5+3i b=3 a=2 b=-3 2 49 a=2 b=7 12 a=0 b= 12 a=7 b=0 2-3i 7 College Algebra Chapter 1.4 Complex Numbers Identify the values of a and b 4 3 25 20 a=0.2 b=0.75 College Algebra Chapter 1.4 Complex Numbers Adding and subtracting complex numbers 4 2i 3 4i 2 3i 10 6i College Algebra Chapter 1.4 Complex Numbers Adding and subtracting complex numbers 5 4i 2 2i 2 3i 5 2i College Algebra Chapter 1.4 Complex Numbers Multiplying Complex Numbers 24 3i 10 i 6 College Algebra Chapter 1.4 Complex Numbers Multiplying Complex Numbers 4 3i 2 i 6 5i 2 4i College Algebra Chapter 1.4 Complex Numbers Multiplying by the Complex Conjugate 2 3i 3 i College Algebra Chapter 1.4 Complex Numbers Division of Complex Numbers 2 5i 6i 7 3i College Algebra Chapter 1.4 Complex Numbers Homework pg 113 1-78 College Algebra Chapter 1.5 Non-factorable Quadratic Equations Standard Form and Coefficients ax bx c 0 x 2 x 2 10 0 z 12 3z 0 2 1 2 x 6x 4 3x 2 9 x 5 2 x 3 0 a=2 b=0 c=-10 a=-3 b=1 c=-12 a=0.25 b=-6 Not quadratic c=0 College Algebra Chapter 1.5 Non-factorable Quadratic Equations Square root property If p 2 k then p k aka p k x 9 2 x 3 or 3 x 3 or p k College Algebra Chapter 1.5 Non-factorable Quadratic Equations solve y 28 0 2 p 36 0 2 College Algebra Chapter 1.5 Non-factorable Quadratic Equations solve 3x 2 28 23 College Algebra Chapter 1.5 Non-factorable Quadratic Equations Completing the Square 1. 2. 3. 4. Subtract the constant c from both sides Divide both sides by the leading coefficient a Take 12 linear coefficien t and add the result to both sides Factor the left hand side as a binomial square: simplify the right hand side 5. Solve using the SQR property of equality 2 College Algebra Chapter 1.5 Non-factorable Quadratic Equations Completing the Square 1. 2. 3. 4. 5. Subtract the constant c from both sides Divide both sides by the leading coefficient a 1 2 linear coefficien t Take and add the result to both sides Factor the left hand side as a binomial square: simplify the right hand side Solve using the SQR property of equality 2 College Algebra Chapter 1.5 Non-factorable Quadratic Equations Quadratic Formula If ax bx c 0 2 Discriminant b b 4ac then x 2a 2 if b2 4ac 0 if b2 4ac 0 1 real root 2 real roots if b2 4ac 0 2 complex roots College Algebra Chapter 1.5 Non-factorable Quadratic Equations Solve Quadratic Formula 9 177 9 177 , 8 8 No Solution College Algebra Chapter 1.5 Non-factorable Quadratic Equations Solve {6,-3} Quadratic Formula {6,1} College Algebra Chapter 1.5 Non-factorable Quadratic Equations Homework pg 126 1-130 College Algebra Chapter 1 Review Solve -10 All Real Numbers No Solution 2 College Algebra Chapter 1 Review Solve and graph, state solution in set and interval notation College Algebra Chapter 1 Review Solve by the method of your choice {5, -5} {2/7, -7} {-4, 6} {7.483, -7.483} {5/8, -8/5} {8, 7} No solution {0.732, -2.732} {1.309, -3.309} {12.426, -4.426} College Algebra Chapter 1 Review Solve {6} {3} {-1} {9, 2} College Algebra Chapter 1 Review Homework pg 129 1-58

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