Answer Key for California State Standards: Algebra I

Algebra I : Symbolic reasoning and calculations with symbols are central in algebra. Through the study of
algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In
addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations

 1.0: Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.

a. Fill in the blanks be low with a single appropriate letter to identify each set of
numbers with the properties or descriptions of the elements which characterize
that set:

The set of:

 E Even Numbers D Rational Numbers G Irrational Numbers C Real Numbers B Integers F Odd Numbers J Natural Numbers H Whole Numbers

A. any number equal to a terminating decimal expression
B. {..., -3, -2, -1, 0, 1, 2, 3, ...}
C. any number which is rational or irrational
D. any number of the form where p and q are integers and q is not zero
E. any integer of the form 2k, where k is an integer
F. any integer of the form 2k + 1, where k is an integer
G. any number equal to an infinite decimal ex pression with no repeating block of
digits
H. {0, 1, 2, 3, ...}
I. any number which can be expressed as a ratio
J. {1, 2, 3, ...}

b.

Which of the following sets of numbers are not closed under addition?

__ The set of real numbers __ The set of rational numbers

The set of irrational numbers __ The set of positive integers

 2.0: Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power . They understand and use the rules of exponents .

a. Which number below is the same as ?

b. What number z satisfies the equation ?

c. Calculate and simplify the expression

d. If x = 4, what is

e. Write as x raised to a power

 3.0: Students solve equations and inequalities involving absolute values.

a |2x - 1| = 5 Find all values of x which make this equation true

 If |2x - 1| = 5, then 2x - 1 = 5 or 2x - 1 = -5 Therefore x = 3 or x = -2

b. 3|2 - 5x| + 1 < 10 Find all values of x which make this inequality true

 Solution: All numbers greater than and less than 1

c. 4|x - 1| = 16 Find all values of x which make this equation true

d. |x- 2| > 4 Find all values for x which make this inequality true

 Either x - 2 > 4 or x - 2 < - 4 Either x > 6 or x < -2 Solution: All numbers less than -2 or greater than 6

 4.0: Students simplify expressions before solving linear equations and inequalities in one variable , such as 3(2x-5) + 4 (x-2) = 12.

a. Simplify

1. 3 ( 2x - 5 ) + 4 ( x - 2 )

b. Solve for x

1. 8(x + 1) + 3(2x - 2) = 44

 5.0: Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

a. Justify each step below for the solution for x from the equation

Use the following list:

A. Commutative Property of Addition
B. Associative Property of Addition
C. Commutative Property of Multiplication
D. Associative Property of Multiplication
E. Distributive Property
F. adding the same quantity to both sides of an equation preserves equality
G. multiplying both sides of an equation by the same number preserves
equality
H. 0 is the additive identity
I. 1 is the multiplicative identity

To the right of each equation below (and on the following pages) where there is an
empty space, write one of the letters 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', or 'I' to justify
how that equation follows from the one above it. For example, the second equation
below is justified by 'G' and the third one by 'E'.

 Step JustificationThe given equation

b. The sum of three integers is 66. The second is 2 more than the
first, and the third is 4 more than twice the first.

What are the integers?

 Let x be the first number. The the second number is x + 2 and the third number is 2x + 4. x + (x + 2) + (2x + 4) = 66 4x + 6 = 66 4x = 60 x = 15 x + 2 = 17 2x + 4 = 34 The three numbers are 15, 17, and 34

c. During an illness, a patient's body temperature T satisfied the inequality
|T - 98.6| ≤ 2. Find the lowest temperature the patient could have had during the
illness.

 |T - 98.6| ≤ 2 -2 ≤ T - 98.6 ≤ 2 -2 + 98.6 ≤ T ≤ 2 + 98.6 96.6 ≤ T ≤ 100.6 The lowest temperature the patient could have had is 96.6

 6.0: Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear equality (e.g., they sketch the region defined by 2x + 6y < 4).

a. Graph the equation: 2x - y = 3

 Two points on the line are (0,-3) and (2,1). Note: Two points determine a line. Calculating the coordinates of any two points on the line determines the same graph.

b. What is the x intercept?

 In the equation 2x - y = 3 (or y = 2x -3), substitute y = 0.

c. What is the y intercept?

 In the equation 2x - y = 3, substitute x = 0 to get y = -3. Alternatively, since b is the y intercept for y = mx + b, it follows that -3 is the y intercept for y = 2x -3.

d. On your graph, mark the region showing 2x - 3 < y

 The shaded region shows 2x - 3 < y
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