Download STEM Name: Practice Set 1 1. Simplify the expression completely

STEM
Practice Set 1
Name: _____________________________
1. Simplify the expression completely: (5x – 7) + 3(2x + 1) – 6(x – 4)
2. The system
3x  y  17
has the solution  2a  1, a  . What is the value of k?
kx  y  18
The interval on a number line starting at 2 and going up to but not including 7 can
be represented in a number of ways.
•
•
•
•
As an inequality: 2 < x < 7
As a graph:
–5
5
0
10
As a set: {x: x > 2 and x < 7} or {x: 2 < x < 7} (This is also sometimes written the a
vertical segment replacing the colon {x| x > 2 and x < 7} or {x| 2 < x < 7})
As an interval: [ 2, 7 )
For each notation there is a way of indicating that the number 2 is included while the
number 7 is not. A solid dot on a graph includes the point, an open circle does not.
A square bracket in interval notation is inclusive the curved parenthesis is not.
3.
a.
Graph: {x: x > –3 and x < 5}
b.
Write 0 < x < 12 in interval notation.
c.
Graph: [6,11)
d.
Write (4, 8] as an inequality
e.
Write 3  x  12 in interval notation
f.
Graph the compound inequality
x   2 or 0  x  3 or x  5
g.
Represent the numbers graphed below in set notation.
–5
0
5
10
15
4. What is the ones digit in the value of each? Be sure you are able to show how you got your answer.
a.
5.
7 43
b.
952
c.
Given the right triangle as shown:
319
Hypotenuse
a
Leg
c
Leg
b
a.
6.
If a = 8 and c = 17, find b
b.
If a = 21 and b = 72, find c
What is the length of BF in the given diagram given AB = AC = CD = DE = EF = 1 and
AB  AC , BC  CD , BD  DE , and BE  EF ?
F
E
B
D
A
7. Find the area of the parallelogram with vertices as (2,1), (9,1), (12,4) and (5,4)
C
Review of Number systems
Natural (counting) numbers
Whole numbers
Integers
Rational numbers
Real numbers
Complex numbers
8.
1, 2, 3, 4, . . .
Natural numbers and zero
0, 1, 2, 3, 4, . . .
Whole numbers and negatives
. . . , -2, -1, 0, 1, 2, . . .
Integers and all fractions and decimals that end or repeat.
All rational numbers can be written as a ratio of two integers.
Rational numbers and irrationals.
Irrational numbers include decimals that do not end or repeat,
and values such as  , and 2 . They are all values
that cannot be written as the ratio of two integers.
Real numbers and imaginary numbers ( 6, 2i , etc.)
List ALL number systems to which each of the following belong. You may use the abbreviations N, W, I,
Q, R, C (in order from above - note that rational numbers are abbreviated as Q - think of “quotient” since
they are the quotient or ratio of two integers.)
7
a.
d.
2
3
b.
2.14114111411114...
c.
6
e.
3  5i
f.
4.153
9. Acute triangle ABC has side AB = 3x  2 , side BC = 4 x  13 and side AC = 2 x  9 . If the perimeter of the
triangle is 174, which is the largest angle of triangle ABC? Why?
10. A number is randomly chosen from the set {1, 2, 3, 4, 5, . . . , 30}.
What is the probability that the number is PRIME? (Remember: 1 is not a prime number.)
11.
The letters from the word SQUARE are put in a bag. A letter is drawn at random from the bag. What is
the probability that the letter:
a.
12.
is a vowel?
b.
comes after M in the alphabet?
Multiply and simplify:
a.
2 x  x  4
b. ( x  3)2  ( x  1)2
2
To evaluate a function such as f ( x)  3x  5 , substitute the value for the variable. For
example, to find f (3) , let x  3 and simplify. So f (3)  3(3)  5 , which gives you
18.f (3)  4 .
You can do this for both numeric values and variable expressions. Using the definition
g ( x)  2 x  7 , evaluating g (3a  1) gives you g (3a  1)  2(3a  1)  7 , so
g (3a  1)  6a  5 .
13. Use the following definitions for the problems below:
3x  1
f ( x)  4 x  2 , g ( x )  x 2  5 , h( x ) 
2x  3
Evaluate:
a.
h(2)
d.
The value of h(2a  3) is undefined. What is the value of a ?
b.
f ( g (2))
c.
f (4k  2)
Factoring and/or Polynomial Multiplication can be used to evaluate algebraic expressions.
For example, given a+b=8 and a 2  b2  50 , you can find the product ab as follows:
a+b=8
then (a+b) 2 =64
Multiplying out, a 2  2ab  b2  64
but a 2  b2  50 , so
2ab + 50 = 64
and ab=7
Another example: given a 2  b2  20 and a+b=5, you can find the difference a-b as follows:
a 2  b2  20
(a+b)(a-b) = 20
but a+b=5, so
5(a-b) = 20
and a-b=4
Note: this is also a quicker way of finding (a,b)
Solving a+b=5, a-b=4 gives (9/2,1/2).
14. a.
x  y  22
xy  40
find the value of x 2  y 2
c.
x  y  10
2 xy  5
find the value of x 2  y 2
15.
b.
x y 8
x y 4
find the value of x 2  y 2
d.
x 2  y 2  30
x y 5
find the value of x  y
If a  b  21 , a  c  40 and b  c  49 , what is the sum a  b  c ? Show how you determined your
answer.
16. Solve for y in terms of a: 5y – 3 = a(3 + 2y)
17. Simplify each expression completely: Give answers as exact values.
a.
75 + 4 48
b.
 2 2   3
40

21

18. Simplify:   6a3b2   a 2b5 
2

19. 3(4x – y) = 5(x + 2y)
20.
Find the ratio of x to y, written in the form x : y.
The shaded area is what percent of the area of the
largest square ?
21.
a. Find the value of the determinant
3 1
4 8
b. Find the value(s) of x so that the determinant
5 3x
is equal to 2.
x x
22. If a  b   a  4  b  1 , find the value of  3   4  6     6  2 
The Arithmetic Mean of n numbers a1 , a2 , a3 , ... an
23.
a1  a2  a3  ...  an
n
is:
The Geometric Mean of n numbers a1 , a2 , a3 , ... an is:
n
(a1 )(a2 )(a3 )  (an ) .
23. Find the exact arithmetic mean and geometric mean of each. Simplify radicals as necessary.
a.
9 and 49
b.
3 5 9
4 ,2 ,5
24. Find the value of x  y where 1  4i 1  4i 1  2i   x  yi
c. 10, 70, 98
25. For each statement, tell if it is ALWAYS, SOMETIMES, or NEVER true.
If it is Always true, give an example (using numeric values) of when it is true.
If it is Sometimes true, give two examples (using numeric values), one when it is true and one when it is
false.
If it is Never true, give an example (using numeric values) of when it is false.
a.
a rational number plus a rational number is a rational number
b.
a rational number times an integer is an integer
c.
the quotient of two rational numbers is a rational number
d.
the product of an irrational number and a rational number is a rational number
e.
the sum of two integers is an integer
f.
the product of two irrational numbers is an irrational number
The distance between two points P1(x1,y1) and P2(x2,y2) is given by the formula:
d
d=
2
(x2 – x1)
2
+ (y2 – y1)
This is based on the Pythagorean Theorem,
using the triangle as shown:
 x2, y1 
26. Find the exact perimeter of triangle FGH if F = (–12,–3) G = (3,5) H = (9,17)
27. The lines y = 2x + 5, y = x + 9, and y = –2x + b intersect at the same point. Find b.
Parallel lines cut transversals proportionally:
a
c
=
b
d
28. Solve for x, y and z.
29. Given set A =  5,  4,  3,  2,  1, 0,1, 2, 3, 4, 5 . If a number is selected at random from set A, what is
the probability that it is a solution to 3x  2 x  3  2  (6 x  1)( x  3) ?
In a fraction containing radicals either the numerator or the denominator can be
rationalized (multiplied by some expression so that the product is a rational number). For
example, to rationalize the denominator:
(1 + 7 )( 5 )
1+ 7
5 + 35
=
=
5
5
( 5 )( 5 )
(multiply by the same radical if there is a single term)
a)
5
b)

1 7

5 1 7

1  7 1  7 

5  35

1  7 
5  35
6
(multiply by the conjugate if there is a binomial)
Note: while it has been standard practice to rationalize the denominator, either a numerator or
a denominator may be rationalized.
Also remember that you should reduce all final fractions to lowest terms.
Examples:
a)
b)
4 2 3
84 3


6
6
30. Simplify:
a.
2
6 2

9
6 2
2 2

3
93


2

4 2 3
  2  2  3  or 4  2
3
63
3
3
3
2
by rationalizing the numerator.
b.
by rationalizing the denominator.
31.
Rationalizing the denominator, giving answer in simplest form:
6
5 3
32. Find all pairs of positive integers (x,y) for which x + 3y = 24. Explain how you got your
answer and how you know you've found all the possibilities.
33. If 5 cows can eat 4 bales of hay in 3 days, how long will it take 6 cows to eat 16 bales of
hay? (Assume all cows eat hay at the same constant rate.) Show your logic.
34. Find the point(s) of intersection of the line y  x  2 and the parabola x 2  3 y  4 .
(Hint: try substitution)
Reminder: One way to solve a quadratic equation (or higher degree) is to set it equal to zero and
use the zero product property. That is, if a • b = 0 then either a = 0 or b = 0.
Example:
Solve x2 + x = 10 – 2x
Organize the equation so that one side is zero.
x2 + 3x – 10 = 0
Factor the quadratic expression
(x + 5)(x – 2) = 0
Then by the zero product property
x+5=0
or
x–2=0
x=–5
x=2
35. Solve each of the following by using the zero-product property. Show your work.
a.
x2 – 4x = 3x + 18
b.
2
x+3
=
x+2
x+8
(Hint: Clear the equation of fractions by
cross-multiplying, or by multiplying
both sides by the common denominator
(x +2)(x + 8) )