College Algebra Test #1 Review, Spring 2010faculty.madisoncollege.edu/kmirus/20092010B/804212/... · Web viewSection 1.1: Linear Equations, Formulas, and Problem Solving Big Idea:

College Algebra Test #01 Review: Equations and Inequalities Page 1 of 16

Section 1.1: Linear Equations, Formulas, and Problem SolvingBig Idea: You can solve any linear equation by combining like terms and using the Properties of Equality to create simpler equivalent equations until you find the solution.Big Skill: You should be able to solve linear equations in one variable, even if there are only letters in the equation!

A. SOLVING LINEAR EQUATIONS USING PROPERTIES OF EQUALITY An equation is a statement that two mathematical expressions are equal. A linear

equation in one variable has only one variable, and the only operations performed on that variable are multiplication by a number or addition. Example:

A common use of equations is to figure out the correct number to put into a calculation to get a desired answer.

Finding the number that satisfies the equation (or makes the equality be true) is called “solving the equation”.

Numbers that make the equation true are called solutions or roots of the equation.

The usual way to solve an equation is to keep re-writing the equation as a sequence of equivalent equations, each one simpler than the one before.

Equivalent equations are two equations that have the same solutions (even if they look different from each other).

Equations can be shown to be equivalent by using the Properties of Equality:

The Additive Property of EqualityIf A, B, and C represent algebraic expressions, and A = B, then A + C = B + C.The Multiplicative Property of EqualityIf A, B, and C represent algebraic expressions, and A = B, then AC = BC, and

(provided that C 0).

Algorithm for Solving Linear Equations in One Variable Eliminate parentheses using the distributive property. (To simplify equations with fractions, you might want to multiply both sides of the

equation by the least common denominator of all the fractions at this point, or even as a first step.)

Combine like terms. Use the Additive Property of Equality to get all variable terms on one side, and all other

terms on the other side. Simplify both sides (by combining like terms). Use the Multiplicative Property of Equality to isolate the variable, as in x = 5. Check your answer in the original equation. For applications, answer the question completely and also include any units of measure.

B. IDENTITIES AND CONTRADICTIONS


A conditional equation is an equation that is only true under the condition that the variable is equal to its solution (or solutions).

An identity is an equation that is always true regardless of the value of the variable. You can tell you have an identity because it will have an equivalent equation like 2 = 2, which is true regardless of the value of the variable (since the variable does not appear in the equation). We write the solution to an identity like: .

A contradiction is an equation that is always false regardless of the value of the variable. You can tell you have a contradiction because it will have an equivalent equation like 2 = -5, which is false regardless of the value of the variable (since the variable does not appear in the equation). We write the solution to a contradiction using the null set or the null symbol: .

C. SOLVING FOR A SPECIFIED VARIABLE IN LITERAL EQUATIONS A literal equation is an equation with two or more variables. Example: 2A2B + CD = B/A

A formula is a type of literal equation in which a quantity is modeled using a mathematical relationship or dependency on one or more other variables. Example: F = ma

It is common to say “solve for B in terms of A, C, and D” to indicate that you are supposed to create a formula from a literal equation by manipulating the equation until you isolate B on one side of the equation and the variables A, C, and D on the other side.

Being able to re-arrange literal equations is useful because then you can derive your own formulas based on given equations. Exciting! Cool! Powerful!

D. USING THE PROBLEM_SOLVING GUIDE Gather and Organize Information

o List given infoo List given formulaso Clarify what you are asked to find

Make the Problem Visualo Draw and label a diagramo Create a table of values

Develop and Equation Modelo Assign a variable to what you are asked to findo Write the equation model, using verbal phrases instead of mathematical

expressions, if necessary Use the Model and Given Information to Solve the Problem

Section 1.2: Linear Inequalities in One Variable


Big Idea: An inequality is used when a range of numbers are desired as the result of a calculation.Big Skill: You should be able to solve linear inequalities using the Properties of Inequalities.

Examples of when you might want to use an inequality: To get an A in this class, you must get between 92% and 100% 0.92 score 1.00 You can only afford a car that costs less than $4,000 price < $4,000 To escape the earth’s gravity, a rocket’s launch speed must be more than 25,000 mph

speed > 25,000 mph

A. INEQUALITIES AND SOLUTION SETS While conditional linear equations in one variable have one solution, linear inequalities

usually have an infinite number of solutions (especially when the answers can be real numbers)

Solution set: the set of all numbers that satisfy an inequality Three ways to represent a solution set:

o Set notationo Interval notation (recall that brackets [] mean to include the endpoints and

parentheses mean ( ) mean that the endpoints are not includedo Number line graph

Comparison of the Representations of an InequalityInequality Set Notation Interval Notation Number Linex is greater than k

x is greater than or equal to k

x is less than k

x is less than or equal to k

x is greater than a and less than b

x is greater than or equal to a and less than bx is greater than or equal to a and less than or equal to bx is greater than a and less than or equal to b

x is less than a or greater than b


x is less than or equal to a or greater than b

x is less than or equal to a or greater than or equal to bx is less than a or greater than or equal to b

B. SOLVING LINEAR INEQUALITIES The usual way to solve a linear inequality is to keep re-writing the inequality as a

sequence of equivalent inequality, each one simpler than the one before. Inequalities can be shown to be equivalent by using the Properties of Inequality:

The Additive Property of InequalityIf A, B, and C represent algebraic expressions, and A < B, then A + C < B + C.Summary: Like quantities can be can be added (or subtracted) to both sides of an inequality without changing the inequality.

The Multiplicative Property of EqualityIf A, B, and C represent algebraic expressions, and A < B, then AC < BC, if C is a positive quantity (i.e., C > 0)

Summary: Like positive quantities can multiply both sides of an inequality without changing the inequality.

If A, B, and C represent algebraic expressions, and A < B, then AC > BC, if C is a negative quantity (i.e., C < 0)

Summary: If like negative quantities multiply both sides of an inequality, then the inequality reverses.

C. SOLVING COMPOUND INEQUALITIES Compound Inequality: when the range of desired solutions must obey more than one

inequality at once.


o Example: If you want to be in the second tax bracket, then you must make $8,350 or more but less than $33,950 income $8,350 and income < $33,950 $8,350 income < $33,950

o Example: If you want to be avoid a parade route, you must be more than 5 blocks east or west of block #22 location > block #22 + 5 or location < block #22 - 5 location > block #27 or location < block #17

Since there is more than one inequality involved in a compound inequality, we will have to figure out how to combine multiple inequalities. This can be done using the notions of set unions and set intersection

Intersection of two sets (i.e., A B): A new set formed that includes only the elements that are common to both sets. This is the first income tax example. Intersections always got with the word “and.”

Union of two sets (i.e., A B): A new set formed that includes all elements that are in either set. This is the second parade example. Unions always got with the word “or.”

D. APPLICATIONS OF INEQUALITIES Writing the domain of functions

o Denominators can not be zeroo The argument to a square root must be greater than or equal to zero

Any real-world problem where a range of values is acceptable for an answer.


Section 1.3: Absolute Value Equations and InequalitiesBig Idea: Absolute values can be thought of as stating the distance from a point. When combined with inequalities, they result in compound inequalities.Big Skill: You should be able to solve absolute value equations and inequalities using the Property of Absolute Value Equations.

Examples of when you might want to use absolute value equations or inequalities: State the two x values that are 5 units away from x = 1

if x is greater than 1, then x – 1 = 5 if x is less than 1, then 1 – x = 5 x – 1 = -5 x – 1 = 5 or x – 1 = -5 |x – 1| = 5

If you want to be avoid a parade route, you must be more than 5 blocks east or west of block #22 location > block #27 or location < block #17 |location – block #22| > 5

A. SOLVING ABSOLUTE VALUE EQUATIONS Isolate the absolute value using the Properties of Equality, then finish the solution using

the Properties of Absolute Values:

Property of Absolute Value EquationsIf X, represents an algebraic expression, and k is a positive real number, then |X| = kimplies X = k or X = -k

Multiplicative Property of Absolute ValueIf A, and B represent algebraic expressions, then |A B| = |A||B|

B. SOLVING “LESS THAN” ABSOLUTE VALUE INEQUALITIES Use the below Property of Absolute Values Inequality:

Property 1 of Absolute Value InequalitiesIf X, represents an algebraic expression, and k is a positive real number, then |X| < kimplies –k < X < k

C. SOLVING “GREATER THAN” ABSOLUTE VALUE INEQUALITIES Use the below Property of Absolute Values Inequality:

Property 2 of Absolute Value InequalitiesIf X, represents an algebraic expression, and k is a positive real number, then |X| > kimplies X < -k or X > kD. APPLICATIONS INVOLVING ABSOLUTE VALUE


Section 1.4: Complex NumbersBig Idea: .Calling the square root of negative one “i” allows us to state the solution of many algebra equations that would otherwise be unsolvable with real numbers.Big Skill: You should be able to perform arithmetic (add, subtract, multiply, and divide) with complex numbers.

A. Identifying and Simplifying Imaginary and Complex NumbersImaginary Numbers and the Imaginary Unit

Imaginary Numbers are those of the form , where k is a positive real number. The imaginary unit i represents the number whose square root is -1:i2 = -1 and

Rewriting imaginary numbers For any positive real number k, .

Complex Numbers Complex numbers are numbers that can be written in the form a + bi, where a and b are

real numbers and is the imaginary unit. The real number a is called the real part of the complex number. The real number b is called the imaginary part of the complex number. The standard form of a complex number is a + bi.

B. Add and Subtract Complex Numbers Adding, subtracting, multiplying, or dividing complex numbers results in an answer that

is also a complex number. To add complex numbers, add “like terms”:

To subtract complex numbers, subtract “like terms”:

C. Multiply Complex Numbers and Find powers of i To multiply complex numbers, use the distributive property, then combine “like terms”

(like using FOIL): One thing to keep in mind: the product property of radicals does not apply when the

radicand is a complex number, so evaluate the square root of the negative number first, then multiply the radicals.

Correct:

Incorrect:


Product of Complex Conjuagtes For a complex number a + bi and its conjugate a – bi, their product (a + bi)( a – bi), is the

real number a2 + b2. To compute powers of i, use an exponent that is the remainder of dividing the original

exponent by 4. This works because the powers of i repeat themselves every four factors of i, as can be seen from making a list of powers of i.

D. Divide Complex Numbers To divide complex numbers, multiply the numerator and denominator by the conjugate

of the denominator, then multiply and simplify. Note: the denominator will always multiply out to a difference of squares, which will be a real number.

Example:


Section 1.5: Solving Quadratic EquationsBig Idea: Quadratic equations can be solved under any circumstance if we allow complex-valued solutions.Big Skill: You should be able to solve any quadratic equation by factoring and using the zero product property, completing the square and using the square root property of equality, and by using the quadratic formula.

A. Solve Quadratic equations Using the Zero Product Property A quadratic equation is an equation in the variable x with constants a, b, and c

(sometimes called parameters) that can be written in the form . The standard form for a quadratic equation is as above, where all non-zero terms are on

the left hand side, and those terms are written in descending powers of the variable.

Quadratic EquationsA quadratic equation can be written in the form with and .

Zero Product PropertyIf A and B represent real numbers or real-valued expressions, and , then .

To solve a quadratic equation using the zero product property:o Get all terms on one side of the equation (which means zero will be on the other

side)o Factor the non-zero sideo Set each factor equal to zero (justified by the zero product property)o Solve each mini “factor equation.”

B. Solve Quadratic equations Using the Square Root Property of EqualitySquare Root Property of EqualityIf X represents an algebraic expression, and , thenwhich is also written as .

To solve a quadratic equation using the square root property of equality:o Manipulate the equation until the left hand side is a perfect square, and there is a

single number on the right hand side.o Apply the square root property of equality.


C. Solve Quadratic Equations by Completing the SquareTo Solve a Quadratic Equation by Completing the Square:(i.e, writing a quadratic trinomial as a perfect square trinomial plus a constant)

Get the constant term on the right hand side of the equation. i.e., if , then write the equation as

Make sure the coefficient of the square term is 1.

Identify the coefficient of the linear term; multiply it by ½ and square the result.

i.e., Find the number b in and compute

Add that number to both sides of the equation.

i.e.,

Write the resulting perfect square trinomial as the square of the binomial .

i.e.,

Use the square root property to solve the equation.

D. Solve Quadratic Equations Using the Quadratic EquationThe Quadratic FormulaThe solution(s) to the quadratic formula (for a 0) are given by the quadratic

formula:

Proof:The quadratic formula is derived by completing the square on the standard from of a quadratic equation:

Get the constant term on the right hand side of the equation.

Make sure the coefficient of the square term is 1.


Identify the coefficient of the linear term; multiply it by ½ and square the result.

Add that number to both sides of the equation.

Write the resulting perfect square trinomial as the square of the binomial .

Use the square root property to solve the equation.


E. Use the Discriminant to Identify Solutions b2 – 4ac is called the discriminant. The discriminant is important because it determines

the nature of the solutions (roots) of the quadratic equation.

Examples of the Nature of the Roots of a Quadratic Equation with Rational Coefficients:1. If b2 – 4ac is positive and a perfect square, then there are two solutions that are real,

rational, and unequal. In this case, you can also solve the quadratic equation by factoring.Example:

2. If b2 – 4ac is positive but not a perfect square, then there are two solutions that are real, irrational, and unequal.Example:

3. If b2 – 4ac = 0, then there is just one solution (a repeated root) that is real and rational (or we can say that the two solutions are equal). This case can also be solved by factoring.Example:

4. If b2 – 4ac is negative, then there are two solutions that are complex and unequal.Example:

F. Solve Applications of Quadratic Equations


Section 1.6: Solving Other Types of EquationsBig Idea: Being able to solve polynomial equations is the key to solving rational, radical, and “quadratic-like” equations.Big Skill: You should be able to solve these four types of algebraic equations.

A. Polynomial Equations of Higher Degree A polynomial equation of degree n is an equation of the form:

Polynomials are the most important topic in algebra because any equation that can be written using addition, subtraction, multiplication, division, integer powers, or roots (which are rational powers) can be solved by converting the equation into a polynomial equation.

For example, the equation can be transformed into the polynomial

equation . Both of these equations have the same solutions, one of which is x = 1, as is shown in the graph below:

Steps for solving any polynomial equation (degree 2 or higher): Expand the polynomial equation (if needed), and collect all terms on one side; combine

like terms. Factor the polynomial on the one side. Set each factor equal to zero (this is justified by the zero-product property rule). Solve each first degree equation for the variable. Check answers in original equation. NOTE: The degree of the polynomial tells how many solutions there must be.



B. Rational EquationsSolving Rational Equations (Book’s Method)

Identify and exclude any values that cause a zero denominator. Multiply both sides by the LCD and simplify. Solve the resulting equation. Check all solutions in the original equation.

Solving Rational Equations (Another Method) Get all terms on one side of the equation. Write all terms as equivalent fractions with a common denominator. Add the terms. Factor the numerator and cancel any common factors in the denominator. Set the numerator to zero and solve the resulting “mini equations”. Check all solutions in the original equation.

C. Radical Equations and Equations with Rational ExponentsSolving a radical equation with one radical expression:

Isolate the radical expression (i.e., get it on one side of the equation by itself with all other terms on the other side).

Raise both sides of the equation to the power of the index of the radical. Solve the resulting equation. Check your answer. It is common to get answers that are not truly solutions. These

answers are called extraneous solutions.

Solving a radical equation with two radical expressions: Isolate ONE of the radical expressions. Raise both sides of the equation to the power of the index of the radical to eliminate the

first radical. Solve the resulting radical equation (that now only has one radical). Check your answer. Eliminate any extraneous solutions.

The Power Property of Equality:If ,

Then

Which simplifies to


D. Equations in Quadratic FormExample:

E. Applications

Documents

College Algebra Test #1 Review, Spring 2010faculty.madisoncollege.edu/kmirus/20092010B/804212/... · Web viewSection 1.1: Linear Equations, Formulas, and Problem Solving Big Idea: