 # Equations and Inequalities

## Presentation on theme: "Equations and Inequalities"— Presentation transcript:

Equations and Inequalities
College Algebra Chapter 1 Equations and Inequalities

Solving equations using
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Solving equations using Addition and multiplication properties of equality

Solving equations using
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Solving equations using Addition and multiplication properties of equality General Approach Simplify the equation Clear fractions or decimals as needed/desired Combine any like terms 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

Solving equations using
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Solving equations using Addition and multiplication properties of equality General Approach Simplify the equation Clear fractions or decimals as needed/desired Combine any like terms 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

Solving equations using
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Solving equations using Addition and multiplication properties of equality General Approach Simplify the equation Clear fractions or decimals as needed/desired Combine any like terms 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

Linear Equation Terminology
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Linear Equation Terminology Algebraic expression Equation Linear equations Standard form Conditional equation Identity Contradictions Sum or difference of algebraic terms Statement that two expressions are equal Three tests – exponents, divisors, multiplication Ax+By=C, where A and B are not both zero Equation is true for one value and false for all others Equation that is always true Equations that are never true

College Algebra Chapter 1
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving No Solution

College Algebra Chapter 1
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving

Linear Equation Terminology
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Linear Equation Terminology Literal equation Formula An equation that has two or more unknowns Equation that models a known relationship between two or more quantities

College Algebra Chapter 1
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Literal Equations:

College Algebra Chapter 1
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Literal Equations:

Consecutive Integer Problems
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
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Mixture Problems

College Algebra Chapter 1
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Mixture Problems

College Algebra Chapter 1
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving

College Algebra Chapter 1
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving

College Algebra Chapter 1
College Algebra Chapter 1.1 Linear Equations, Formulas, and Problem Solving Homework pg

Inequalities and Solution Sets Terminology
College Algebra Chapter 1.2 Linear Inequalities in One Variable Inequalities and Solution Sets Terminology Solution Set Set Notation Number Line Interval Notation Boundary Point Inclusion Not Included The set of numbers that satisfy an inequality Method of writing a solution set. Ex {x|x>k} Method of writing a solution set Method of writing a solution set. Ex Marks the location of a number on a number line also called an endpoint Uses brackets [ ] to indicate Uses parenthesis ( ) to indicate

Inequalities and Solution Sets Terminology
College Algebra Chapter 1.2 Linear Inequalities in One Variable Inequalities and Solution Sets Terminology Set Notation The set of all n Such that n is greater than or equal to 1

Inequalities and Solution Sets Terminology
College Algebra Chapter 1.2 Linear Inequalities in One Variable Inequalities and Solution Sets Terminology Interval Notation 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

Write the following in set notation, number line graph,
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 Number line graph 10 Interval notation

Write the following in set notation, number line graph,
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 Number line graph 2 Interval notation

Write the following in set notation, number line graph,
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 Number line graph -1 5 Interval notation

Solving Inequalities Beware of multiplying or dividing by a (-)
College Algebra Chapter 1.2 Linear Inequalities in One Variable Solving Inequalities Beware of multiplying or dividing by a (-)

College Algebra Chapter 1.2 Linear Inequalities in One Variable
Solving Inequalities

Solving Compound Inequalities
College Algebra Chapter 1.2 Linear Inequalities in One Variable Solving Compound Inequalities

Solving Compound Inequalities
College Algebra Chapter 1.2 Linear Inequalities in One Variable Solving Compound Inequalities

College Algebra Chapter 1.2 Linear Inequalities in One Variable
Homework pg

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 , then either

Remove common factors aka undistribute
College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solve Set equal to zero Remove common factors aka undistribute Factor trinomial Solve each factor equal to zero

College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Solve

College Algebra Chapter 1.3 Solving Polynomial and Other Equations

Solving rational equations
College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 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

Solving rational equations
College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 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

Solving rational equations
College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 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

Solving rational equations
College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 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

Solving rational equations
College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving rational equations 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 Isolate radical term Square both sides Simplify

College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations Isolate radical term Square both sides Simplify 3

College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations Isolate radical term Square both sides Simplify

College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations Isolate radical term Square both sides Simplify

College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving Radical Equations Isolate radical term Square both sides Simplify

College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Rational Exponents 15

College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Rational Exponents

Solving with U-Substitution
College Algebra Chapter 1.3 Solving Polynomial and Other Equations Solving with U-Substitution

College Algebra Chapter 1.3 Solving Polynomial and Other Equations
Homework pg

Imaginary unit i represents the number whose square is -1
College Algebra Chapter 1.4 Complex Numbers Imaginary unit i represents the number whose square is -1

College Algebra Chapter 1.4 Complex Numbers

College Algebra Chapter 1.4 Complex Numbers
Complex numbers are those that can be written in the form , where a and b are real numbers and The expression is called the standard form of a complex numbers Real part Imaginary part

Identify the values of a and b
College Algebra Chapter 1.4 Complex Numbers Identify the values of a and b a=-5 b=3 -5+3i 2-3i a=2 b=-3 a=2 b=7 a=0 b= a=7 b=0 7

Identify the values of a and b
College Algebra Chapter 1.4 Complex Numbers Identify the values of a and b a=0.2 b=0.75

College Algebra Chapter 1.4 Complex Numbers Adding and subtracting complex numbers

College Algebra Chapter 1.4 Complex Numbers Adding and subtracting complex numbers

Multiplying Complex Numbers
College Algebra Chapter 1.4 Complex Numbers Multiplying Complex Numbers

Multiplying Complex Numbers
College Algebra Chapter 1.4 Complex Numbers Multiplying Complex Numbers

Multiplying by the Complex Conjugate
College Algebra Chapter 1.4 Complex Numbers Multiplying by the Complex Conjugate

Division of Complex Numbers
College Algebra Chapter 1.4 Complex Numbers Division of Complex Numbers

College Algebra Chapter 1.4 Complex Numbers
Homework pg

Standard Form and Coefficients
College Algebra Chapter 1.5 Non-factorable Quadratic Equations Standard Form and Coefficients a=2 b=0 c=-10 a=-3 b=1 c=-12 a= b= c=0 Not quadratic

College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Square root property

College Algebra Chapter 1.5 Non-factorable Quadratic Equations
solve

College Algebra Chapter 1.5 Non-factorable Quadratic Equations
solve

Completing the Square Subtract the constant c from both sides
College Algebra Chapter 1.5 Non-factorable Quadratic Equations Completing the Square Subtract the constant c from both sides Divide both sides by the leading coefficient a 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

College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Completing the Square Subtract the constant c from both sides Divide both sides by the leading coefficient a 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

Quadratic Formula Discriminant 1 real root 2 real roots
College Algebra Chapter 1.5 Non-factorable Quadratic Equations Quadratic Formula Discriminant 1 real root 2 real roots 2 complex roots

College Algebra Chapter 1.5 Non-factorable Quadratic Equations Quadratic Formula Solve No Solution

College Algebra Chapter 1.5 Non-factorable Quadratic Equations Quadratic Formula Solve {6,-3} {6,1}

College Algebra Chapter 1.5 Non-factorable Quadratic Equations
Homework pg

Solve -10 No Solution All Real Numbers 2
College Algebra Chapter 1 Review Solve -10 No Solution All Real Numbers 2

Solve and graph, state solution in set and interval notation
College Algebra Chapter 1 Review Solve and graph, state solution in set and interval notation

Solve by the method of your choice
College Algebra Chapter 1 Review Solve by the method of your choice {5, -5} {7.483, } {5/8, -8/5} {2/7, -7} {8, 7} {-4, 6} {0.732, } No solution {1.309, } {12.426, }

College Algebra Chapter 1 Review
Solve {6} {3} {-1} {9, 2}

College Algebra Chapter 1 Review
Homework pg