Chapter 1

Foundations

Read chapter 1 in the book

Summary

Put commas in the number to separate the periods Draw 3 blanks to indicate the number of places needed in each period. Name the number in each period and place the digits in the correct place value position Locate the given place value and mark it with an arrow.

Key terms

absolute value

The absolute value of a number is its distance from 0 on the number line. The absolute value of a number n is written as | n |

denominator

The denominator is the value on the bottom part of the fraction that indicates the number of equal parts into which the whole has been divided

fraction

A fraction is written a b , where b ≠ 0 a is the numerator and b is the denominator. A fraction represents parts of a whole. The denominator b is the number of equal parts the…

irrational number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat

number line

A number line is used to visualize numbers. The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left

opposite

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero: - a means the opposite of the number. The notation…

rational number

A rational number is a number of the form p q , where p and q are integers and q ≠ 0 . A rational number can be written as the ratio of two integers. Its decimal form stops or…

Chapter 2

Solving Linear Equations and Inequalities

Read chapter 2 in the book

Summary

To Determine Whether a Number is a Solution to an Equation Step 1. Substitute the number in for the variable in the equation If it is not true, the number is not a solution

Key terms

Number

a Solution to an Equation Step 1

conditional equation

An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation

identity

An equation that is true for any value of the variable is called an identity. The solution of an identity is all real numbers

contradiction

An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution

solution of an equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation

Chapter 3

Math Models

Read chapter 3 in the book

Summary

Make sure all the words and ideas are understood Choose a variable to represent that quantity It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation

Key terms

principal

The principal is the original amount of money invested or borrowed for a period of time at a specific interest rate

amount of discount

The amount of discount is the amount resulting when a discount rate is multiplied by the original price of an item

discount rate

The discount rate is the percent used to determine the amount of a discount, common in retail settings

rate of interest

The rate of interest is a percent of the principal, usually expressed as a percent per year

interest

Interest is the money that a bank pays its customers for keeping their money in the bank

mark-up

A mark-up is a percentage of the original cost used to increase the price of an item

original cost

The original cost in a retail setting, is the price that a retailer pays for an item

simple interest

Simple interest is the interest earned according to the formula I = P r t

Chapter 4

Graphs

Read chapter 4 in the book

Summary

Solution of a Linear Equation An ordered pair ( x , y ) is a solution of the linear equation A x + B y = C , if the equation is a true statement when the x - and y - values of the ordered pair are substituted into the… Points with a y -coordinate equal to 0 are on the x -axis, and have coordinates ( a , 0 ) Sign Patterns of the Quadrants Quadrant I Quadrant II Quadrant III Quadrant IV ( x , y ) ( + , + ) ( - , - ) ( + , - ) Quadrant I Quadrant II Quadrant III Quadrant IV ( x , y ) ( + , + ) ( - , - ) ( + , - ) Points with an x -coordinate equal to 0 are on the y -axis, and have coordinates ( 0 , b )

Key terms

linear equation

A linear equation is of the form A x + B y = C , where A and B are not both zero, is called a linear equation in two variables

rectangular coordinate system

A grid system is used in algebra to show a relationship between two variables; also called the xy -plane or the ‘coordinate plane’

equation

a true statement when the x - and y - values of the ordered pair are substituted into the…

ordered pair

An ordered pair ( x , y ) gives the coordinates of a point in a rectangular coordinate system

quadrant

The x -axis and the y -axis divide a plane into four regions, called quadrants

y -coordinate

The second number in an ordered pair ( x , y )

y -coordinate equal to 0

on the x -axis, and have coordinates ( a , 0 )

x -coordinate equal to 0

on the y -axis, and have coordinates ( 0 , b )

Chapter 5

Systems of Linear Equations

Read chapter 5 in the book

Summary

To solve a system of linear equations by graphing Step 1. Determine whether the lines intersect, are parallel, or are the same line If the lines intersect, identify the point of intersection. If the lines are parallel, the system has no solution.

Key terms

system of linear equations

When two or more linear equations are grouped together, they form a system of linear equations

lines

parallel, the system has no solution

dependent equations

Two equations are dependent if all the solutions of one equation are also solutions of the other equation

supplementary angles

Two angles are supplementary if the sum of the measures of their angles is 180 degrees

complementary angles

Two angles are complementary if the sum of the measures of their angles is 90 degrees

consistent system

A consistent system of equations is a system of equations with at least one solution

inconsistent system

An inconsistent system of equations is a system of equations with no solution

Chapter 6

Polynomials

Read chapter 6 in the book

Summary

Monomial —A polynomial with exactly one term is called a monomial Trinomial —A polynomial with exactly three terms is called a trinomial Degree of a Polynomial The degree of a term is the sum of the exponents of its variables Polynomials polynomial —A monomial, or two or more monomials combined by addition or subtraction is a polynomial

Key terms

polynomial

A polynomial is a monomial, or two or more monomials combined by addition or subtraction

monomial

A monomial is a term of the form a x m , where a is a constant and m is a whole number; a monomial has exactly one term

degree of a polynomial

The degree of a polynomial is the highest degree of all its terms

degree of a term

The degree of a term is the exponent of its variable

trinomial

A trinomial is a polynomial with exactly three terms

binomial

A binomial is a polynomial with exactly two terms

Chapter 7

Factoring

Read chapter 7 in the book

Summary

Finding the Greatest Common Factor (GCF): To find the GCF of two expressions: Step 1. Write all variables with exponents in expanded form List all factors—matching common factors in a column.

Key terms

factoring

Factoring is splitting a product into factors; in other words, it is the reverse process of multiplying

greatest common factor

The greatest common factor is the largest expression that is a factor of two or more expressions is the greatest common factor (GCF)

perfect square trinomials pattern

If a and b are real numbers, a 2 + 2 a b + b 2 = ( a + b ) 2 a 2 - 2 a b + b 2 = ( a - b ) 2 a 2 + 2 a b + b 2 = ( a + b ) 2 a 2 - 2 a b + b 2 = ( a - b ) 2

Zero Product Property

The Zero Product Property states that, if the product of two quantities is zero, at least one of the quantities is zero

prime polynomials

Polynomials that cannot be factored are prime polynomials

quadratic equations

are equations in which the variable is squared

difference of squares pattern

If a and b are real numbers,

Chapter 8

Rational Expressions and Equations

Read chapter 8 in the book

Summary

Simplified Rational Expression A rational expression is considered simplified if there are no common factors in its numerator and denominator Opposites in a Rational Expression The opposite of a - b is b - a . Multiplication of Rational Expressions If p , q , r , s are polynomials where q ≠ 0 , s ≠ 0 , then p q · r s = p r q s To multiply rational expressions, multiply the numerators and multiply the denominators

Key terms

rational expression

A rational expression is an expression of the form p q , where p and q are polynomials and q ≠ 0

complex rational expression

A complex rational expression is a rational expression in which the numerator or denominator contains a rational expression

proportion

A proportion is an equation of the form a b = c d , where b ≠ 0 , d ≠ 0 . The proportion is read “ a is to b , as c is to d . ”

similar figures

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio

rational equation

A rational equation is two rational expressions connected by an equal sign

Chapter 9

Roots and Radicals

Read chapter 9 in the book

Summary

If a n is a real number and n ≥ 2 , a 1 n = a n For any positive integers m and n , a m n = ( a n ) m and a m n = a m n

Key terms

rational exponents

If a n is a real number and n ≥ 2 , a 1 n = a n . For any positive integers m and n , a m n = ( a n ) m and a m n = a m n

rationalizing the denominator

The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator

radical equation

An equation in which the variable is in the radicand of a square root is called a radical equation

like radicals

Radicals with the same index and same radicand are called like radicals

like square roots

Square roots with the same radicand are called like square roots

principal n th root

The principal n th root of a is written a n

square root of a number

If n 2 = m , then n is a square root of m

Chapter 10

Quadratic Equations

Read chapter 10 in the book

Summary

Square Root Property If x 2 = k , and k ≥ 0 , then x = k or x = - k Quadratic Formula The solutions to a quadratic equation of the form a x 2 + b x + c = 0 , a ≠ 0 are given by the formula: x = - b ± b 2 - 4 a c 2 a Solve a Quadratic Equation Using the Quadratic Formula To solve a quadratic equation using the Quadratic Formula. Binomial Squares Pattern If a , b are real numbers, ( a + b ) 2 = a 2 + 2 a b + b 2 ( a - b ) 2 = a 2 - 2 a b + b 2

Key terms

quadratic equation

A quadratic equation is an equation of the form a x 2 + b x + c = 0 , where a ≠ 0

Square Root Property

The Square Root Property states that, if x 2 = k and k ≥ 0 , then x = k or x = - k

consecutive even integers

Consecutive even integers are even integers that follow right after one another. If an even integer is represented by n , the next consecutive even integer is n + 2 , and the…

consecutive odd integers

Consecutive odd integers are odd integers that follow right after one another. If an odd integer is represented by n , the next consecutive odd integer is n + 2 , and the next…

quadratic equation in two variables

A quadratic equation in two variables, where a , b , and c are real numbers and a ≠ 0 is an equation of the form y = a x 2 + b x + c

vertex

The point on the parabola that is on the axis of symmetry is called the vertex of the parabola; it is the lowest or highest point on the parabola, depending on whether the…

axis of symmetry

The axis of symmetry is the vertical line passing through the middle of the parabola y = a x 2 + b x + c

discriminant

In the Quadratic Formula, x = - b ± b 2 - 4 a c 2 a the quantity b 2 - 4 a c is called the discriminant