• The sections in the Worktext do NOT follow a mathematical decimal pattern.  They follow a chapter - section pattern.  So, 1.12 means "chapter 1, section 12" and comes after 1.11.  In other words the order is 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, 1.12, and so on.
• The Worktext Section below is linked to a website with explanations and exercises.  You will want to explore the options on this site to both see what is available and to help you learn.

....An important idea is "order of operations" 

• This is a site I have written.
• This one is from math.com.  I like that it includes explanations and examples for you to work.  However, it does have annoying banners.
• And another one from purplemath.

...Another basic concept is "using the distributive property and combining like terms"

• This is a site I have written.
• Purple Math at http://www.purplemath.com/modules/polyadd.htm gives additional explanations.
• Ask Dr. Math at http://mathforum.org/library/drmath/view/56380.html gives another explanation.

• For a more complete explanation with sample problems go to http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=polyadd.

..This site will help you understand the relationship among sets of numbers

• This is a site I have written.

..An important skill set is the ability to solve linear equations

• This is a site I have written.
• Purple Math at http://www.purplemath.com/modules/solvelin.htm gives additional explanations.
• For a more complete explanation with sample problems go to http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=solveq

..To help you solve inequalities

• This is a site I have written.

Beginning Algebra

119 Objectives

Sample Problem

Worktext Section

I. Real Numbers

A. Operations, substitution and evaluation

alge005: Evaluation of a linear expression in two variables

Evaluate the expression
x –  3c when x = 2 and c = – 6.

1.3

alge004: Evaluation of a polynomial in one variable

Evaluate the expression when c = – 2

1.4

B. Algebraic symbols

alge015: Writing an inequality

Write the algebraic expression for the sentence
" 3 is greater than a.”

1.6

alge016: Translating sentences into linear equations

Write the algebraic expression for the sentence
" 3 plus a equals 2."

1.5

alge186: Writing a compound inequality

Write " x is greater than or equal to – 6 and less than – 4" as an algebraic expression.

1.7

alge602: Writing a mathematical expression

Write an algebraic expression to answer the question below.
How many minutes are there in h hours?
Answer:  60h minutes

C. Number system

alge001: Integer and rational numbers

For each of the two number systems, indicate all the numbers which belong to it. (Problems in ALEKS)

1.1

alge002: Integer, rational and irrational numbers (real)

For each of the four number systems, indicate all the numbers which belong to it.
(Problems in ALEKS)

1.2

D. Properties of real numbers

alge606: Distributive Property, simple

Use the Distributive Property to remove the parentheses from the following expression:
(u – 12)(5)
Answer: 5u – 60

1.9

alge604: Distributive Property, advanced

Use the Distributive Property to remove the parentheses from the following expression:
(4 – 3y + 4t)(-7))
Answer: – 28 + 21y  – 28t

1.9

alge607: Combining like terms, basic

Simplify the following expression:
-12w + 3 w
Answer: - 9w

1.9

alge187: Properties of addition

For each equation below, indicate the property that justifies the equation by filling in the box with the appropriate number.

   6 + (b + 4) = (6 + b) + 4     [   ]
   0 = (-3) + 3                         [   ]
   5 + 0 = 5                            [   ]

Answer:

   6 + (b + 4) = (6 + b) + 4     [ 2 ]
   0 = (-3) + 3                         [ 4 ]
   5 + 0 = 5                            [ 3 ]

1.9

alge188: Properties of real numbers

For each equation below, indicate the property that justifies the equation by filling in the box with the appropriate number

  9 + 3 = 3 + 9                 [   ]

                         [   ]

   5 + 0 = 5                      [   ]

           [   ]

Answer:

  9 + 3 = 3 + 9                 [ 1 ]

                         [ 9 ]

   5 + 0 = 5                      [ 3 ]

           [ 7 ]

1.9

II. Solving Linear Equations

A. One occurrence of the variable

alge010: Additive property of equality: Problem type 2

What is the value of  w if
w + 2 = – 9?

2.2

alge226: Additive property of equality: Problem type 3

If 153 = 242 –  w,
what is the value of w?

2.3

alge802:Multiplicative property of equality with fractions

alge008:Multiplicative property of equality: Problem type 1

Solve for w: 3w = 87

2.4

alge012: Multiplicative property of equality: Problem type 2

Solve the following equation for t:  

2.5

alge006: Solving a linear equation: Problem type 1

Solve the following equation for t:
 2t – 3 = 5

2.6

alge208: Solving a linear equation: Problem type 2

Solve for x:  

2.7

alge200: Solving a linear equation: Problem type 3

Evaluate the expression
2x + 10, given that 11x – 2 = 9.

2.8

B. Several occurrences of the variable

alge011: Solving a linear equation with several occurrences of the variable: Problem type 1 

What is the value of u if 
2(u– 4) – 5u = – 29

2.9

alge061: Solving a linear equation with several occurrences of the variable: Problem type 2 

Solve for x:  Write your answer as a fraction in simplest form.

Answer:

2.10

alge013: Solving a linear equation with several occurrences of the variable: Problem type 3

Solve the following equation for x :

2.11

alge209: Solving a linear equation with several occurrences of the variable: Problem type 4

Solve the following equation for x:

2.12

alge179: Solving a linear equation with several occurrences of the variable: Problem type 5

Solve for u:

2.13

The surface area S of a right prism is given by

S = 2 B + Ph,
where B is the area of the base, P is the perimeter, and h is the height of the prism. Solve for P.

5.17

C. Inequalities

alge019: Solving a linear inequality: Problem type 1

Solve the inequality t – 5 <  20 for t

2.16

alge020: Solving a linear inequality: Problem type 2

Solve the inequality
11x – 2 > 9 for x.

2.17

alge021: Solving a linear inequality: Problem type 3

Solve the inequality
– 4z – 19 < – 3 for z.

2.18

alge207: Solving a linear inequality: Problem type 4

Solve the inequality  for w.

2.19

alge017: Graphing a linear inequality on the number line

2.20

alge166: Graphing a compound linear inequality on the number line

Graph the portion of the number line containing all points for which
x < – 6 or x > 9

2.21

D. Applications

alge014: Solving a word problem using a linear equation: Problem type 1

A total of 328 tickets were sold for the school play. The number of student tickets sold is three times the number of adult tickets sold. How many student tickets were sold?
Answer:
Let S = number of student tickets sold, and
 and A = number of adult tickets sold.
Since the number of student tickets sold is three times the number of adult tickets sold S=3A.
Since, S = 3A and S + A = 328,
3A + A = 328.
Solving for A we have, A = 82.
S = 3A = 246 tickets.
There were 246 adult tickets sold.

2.14

alge219: Solving a word problem using a linear equation: Problem type 2

2.15

Two discounted bouquets of flowers were sold for $43.34. If the discount on each bouquet of flowers was $9.28, what was the price of each bouquet of flowers before discount? (Suppose that each bouquet of flowers has the same price).

2.29

alge704: Solving a word problem using a linear equation: Problem type 4

alge022: Word problem with linear inequalities

The sum of two numbers is less than 12. The second number is 6 more than the first. What are the possible values for the first of the two numbers?
In your answer, denote the first number by x.
Answer:
Let x = the first number
   x + 6 = second number
  x +  x + 6 < 12
  2x  + 6 < 12
  2x < 12
  x < 6

2.22

mstat014: Random samples and prediction

In a magazine survey, 150 randomly selected students from several middle schools throughout the United States are asked whether they have access to a computer at home and if they have a personal e-mail account. The responses of the students are summarized in the following table:

(a)  What percentage of the students surveyed have an e-mail account?
(b)  Answer: 57/150 = 0.38 = 38%

What percentage of the students surveyed have no computer access at home?
Answer: 84/150 = 0.56 = 56%

geom078: Sides of polygons having the same perimeter

Afterward, the wire is bent into an equilateral triangle. Thus, the perimeter of the equilateral triangle is also 48cm.
That is, if S is the length of a side of the equilateral triangle, we have
3S = 48
S = 16 cm

geom143: Area and perimeter of a rectangle

Since the dimension of a rectangle cannot be negative we reject W = -6, so W = 6 ft, and L = 2W = 12 ft.

The perimeter = 2L + 2 W = 2(6) + 2(12) = 12 + 24 = 36 ft.

The perimeter is 36 ft.

alge103: Solving a linear equation with absolute value: Basic

List all the solutions of the equation |4t + 6| = 22.

2.23

III. Graphs and Functions

A. Ordered pairs

alge066: Solutions to a linear equation in two variables: Problem type 1

Provide a pair of values for
(t, x) that satisfies the equation
2t – x = 9.

3.4

alge064: Reading a point in the coordinate plane

Give the location of Rome as an ordered pair (x,y).
(Problems in ALEKS)

3.1

alge067: Plotting a point in the coordinate plane

Using the pencil, mark a point at the coordinates (– 6, 9)

3.2

alge216: Solutions to a linear equation in two variables: Problem type 2

Indicate, by checking the appropriate box, whether the given points are on line 1, on line 2, on both line 1 and line 2, or on neither one of the two lines:
Line 1: 4x + 3y + 17 = 0
Line 2: – 4x + 5y + 7 = 0

3.5

B. Graphing lines

alge197: Graphing a line given the x and y intercepts

Graph a line that intercepts the y-axis at 9, and the x-axis at
– 6.

3.6

alge194: Graphing a line given the equation in slope-intercept form

Graph the line y = – 2x – 9

3.7

Graph the line 9x + 8y – 11 = 0

3.8

alge198: Graphing a vertical or horizontal line

Graph the line y = – 6

3.10

C. Equations of lines

alge210: X and Y intercepts of a line given the equation in standard form

Find both the x-intercept and the y-intercept of the line given by the equation
2x – 9y + 15 = 0

3.12

E. Sets

set001: Set builder notation

Rewrite the set L by listing its elements.  Make sure to use the appropriate set notation: 
Answer:

We must list all values of z that satisfy the following three conditions.

   1. z is an integer

   2. z is greater than or equal to -5

   3. z is less than or equal to -2.

The answer is, L = {-5, -4, -3, -2}

Set002: Union and intersection of sets

The sets E and H are given below.

What is the union of the sets E and H? The symbol for union looks like a “U” and is
What is the intersection of the sets E and H?  The symbol for intersection is .

Answer:
E = { -3, -2, -1, 0 1}    H = { -4, -3, -2, -1}

F. Functions

fun001: Function tables

Answer:

fun001: Graphing integer functions

Answer:

IV. Systems of Linear Equations

alge078: Word problem on systems of linear equations: Problem type 1

Find two numbers whose sum is 69, and whose difference is 11.

2.33

V.  Exponents and Polynomials

A. Properties of exponents

alge024: Product rule of exponents  (Additive Laws of Exponents)

Express the following product of powers of u as a single power of u:

5.4

alge026: Substitution and laws of exponents: Problem type 1

If , then w?

5.6

alge027: Substitution and laws of exponents: Problem type 2

Given that , rewrite
 in terms of z only and simplify as much as possible.

5.7

arith042: Writing a positive number without a negative exponent

Rewrite the following without an exponent:
  
Answer:
  

Arith 6.9

arith043: Writing a negative number without a negative exponent

Rewrite the following without an exponent:
  
Answer:
  

Arith 6.11

alge025: Quotient of exponents and negative exponents  (Multiplication Law of Exponents and Negative Exponents)

Simplify: and write your answer in such a way as to avoid negative exponents.

5.5

alge028: Product rule of exponents in a multivariate monomial  (Multiplication Law of Exponents and Negative Exponents)

Simplify  as much as possible.

5.5

arith029: Ordering numbers with positive exponents

For each of the three expressions, decide which side is greater or whether the two sides are equal. Then fill in each box with one of the following symbols: >(greater than),< (less than), or =(equal).
  

Answer:

Arith 6.1

arith024: Ordering numbers with negative exponents

Make each of the following three mathematical expressions correct by filling in the boxes with the one of these symbols: >(greater than),< (less than), or =(equal).
  

Answer:

Arith 6.10

B. Scientific notation

arith036: Scientific notation with positive exponent

Write 17,000 in scientific notation.
Answer:

Arith 6.12

arith037: Scientific notation with negative exponent

Write 0.000017 in scientific notation.
Answer:

Arith 6.13

C. Manipulating polynomial expressions

alge031: Degree of a multivariate polynomial

What is the degree of the polynomial:

6.6

alge029: Simplifying a polynomial expression

Add the following three polynomials:
Simplify the result as much as possible.

6.1

alge033: Multiplication of two binomials

Express the following product as a polynomial in w:
(w + 2)(w – 8)

6.3

alge030: Multiplication of monomials

Perform the following multiplications:

and simplify your answer as much as possible

6.2

alge032: Squaring a binomial

Expand the square

6.4

alge180: Multiplication of polynomials

Multiply
(2z + 7x + 6)(6z + 2x – 2)
and simplify as much as possible.

6.5

D. Factoring

alge039: Factoring a quadratic with leading coefficient 1

Factor

Answer:

6.13

alge040: Factoring a quadratic with leading coefficient greater than 1

Factor

Answer:

6.14

alge043: Factoring a perfect square

Write  as a perfect square

6.15

alge037: Greatest common factor of two monomials

Find the greatest common factor of the two expressions
and    Simplify your answer 

 as much as possible.

6.9

alge038: Factoring a multivariate polynomial by grouping: Problem type 1

Factor:

 

6.11

VI. Rational expressions and functions

A. Simplifying expressions

alge053: Multiplying rational expressions

Multiply:  and reduce your answer to lowest

 terms.

5.9

alge056: Adding rational expressions with common denominator

Express the following sum as a single fraction:
.  Reduce the result to lowest terms

 whenever possible.

5.13

B. Solving equations

alge049: Restriction on variable in a denominator

If the expression:
 is to make sense, the values that u can take must be restricted. State the restriction on u

5.1

alge060: Solving a rational equation that simplifies to a linear equation: Problem type 1

Solve for

2/u = 8.

Simplify your answer as much as possible.
Answer:

2=8u

1/4=u

5.18

E. Direct and inverse variations

alge059: Ordering fractions

Suppose a, b, c, and d are four numbers which satisfy the conditions a > b > c > d > 0.
Fill in the blank boxes using either >, <, or = to make each of the following statements true.

Hint:  The easiest way to do these problems is to replace the variables with numbers that work in the original inequality and then check what happens in each statement.

Answer:

5.2

alge220: Word problem on proportions: Problem type 3

3 machines can complete a given task in 8 days. If there are 4 machines, how many days would it take for them to finish the same task?

5.3

geom037: Similar polygons

The pentagons ABCDE and PQRST are similar.  Find the length x or the segment . 

Answer:

There are many proportions that can be made.  Any two corresponding sides have equal ratios.

VII. Radicals and rational exponents

Rewrite the following in simplified radical form.

Answer:

First find the prime factorization of 72:

Remember that when you take the square root you are undoing squaring.  So the sqrt (2 x 2) = 2, and so on.

Simplify the following expression as much as possible.

Answer:

Find the prime factorization of both numbers inside square root signs and take square roots.  (see above)

geom44: Pythagorean Theorem

For the following right triangle, find the side length x. Round your answer to the nearest tenth.

Answer:

Compute the following:

Answer: