Download STEM Name: Practice Set 2 1. 2(x + 10y) = 5(4x + 2y) Find the ratio y

STEM
Practice Set 2
Name:____________________________________
1.
2(x + 10y) = 5(4x + 2y)
2.
How many different 3 X 3 matrices are there that use the integers 1 through 9
a.
Find the ratio y : x in simplified form where y and x are both integers.
once each?
b.
with repeats allowed?
3.
7
4x-1
13
30
The perimeter of the square is 40% of the perimeter of the triangle. Determine the value of x.
4.
Find the measures of both angles in this linear
pair.
5. The slope of the line through the points (5, − 7 + a ) and (9, 11 + 3a ) is 6.
Find the value of a.
6. Simplify:
7.
(3–
5)
2
+ 180
x2 = x is not always true. Find two real values of x for which it is false.
8.
Find the value of y:
Triangle Inequality Theorem:
Remember: the shortest distance between two points is a straight line. Therefore, the distance from A to B (z)
must be less than the distance from A to C (y) plus the distance from C to B (x).
That is: x + y > z
Similarly: x + z > y or, solving for z, z > y – x (if y is greater than x)
In short: the length of the third side of a triangle must be greater than the absolute value of the difference of
the other two sides and less than the sum of the other two sides. Algebraically this is written as: | x-y | < z <
x+y (similar inequality for all three sides) This is often referred to as the Triangle Inequality Theorem.
9.
a. The length of side y in the triangle must be
between two values in order for the
triangle to exist. What are these values?
Give answer as an inequality:
min. length < y < max. length
b. The third side of a triangle with sides of integral lengths is described by the following
inequality: 4 < x < 14. Find the lengths of the other two sides.
While the zero product property works well as a method for solving many quadratic
equations, other methods are also possible.
If a perfect square trinomial is equal to a number, taking the square root of both sides of the equation is a
good way to proceed.
For example:
then
If (x + 4)2 =
|x+4 |
x+4
25
=
= ±5
x
=
–4 ± 5
x
=
–9 or 1
If 9x2 – 6x + 1 =
7
then (3x – 1)2
7
| 3x – 1 |
3x – 1
25
=
=
7
=
± 7
3x
=
1± 7
x
=
1± 7
1
7
or
± 3
3
3
10. Solve for x using the square root method, giving answers as exact values. Show your work
a.
(x – 5)2 = 14
b.
4x2 + 12x + 9 = 81
11. If A = lwh and if l, w, and h are all doubled, how will the value of A be changed?
A factorial (indicated by an exclamation point "!") is defined for positive integers as the product of all
whole numbers between the integer and 1, inclusive.
0! is defined to equal 1.
Factorials are not defined for negative and or fractional/decimal/irrational values.
12.
a. Find n if n! = 40,320
b. Simplify:
54! 16!
+
53! 14!
(Without using the factorial key on your calculator. Be able to show your work.)
13. A collection of 30 coins worth $5.50 consists of nickels, dimes, and quarters. There are twice
as many dimes as nickels. How many quarters are there?
14. Given that x +
1
1
=
3 , find the value of x 2 + 2
x
x
15. Harry wants to be fair in dividing up the candy he brought to school between Ron and Hermoine. 1
are the same as
1
Bertie beans
2
3
of a chocolate frog. How many chocolate frogs would equal 12 Bertie beans?
4
16. Given f ( x )= x + 3 , solve for a so that f (a ) =
f (1) ( f (7) + f (2) )
f (3)
Negative Exponents
x8
1
xa
If we reduce the expression 10 we get 2 . However, if we apply the rule b = xa–b
x
x
x
1
we get x–2 . Since both answers, 2 and x–2 , came from the same problem, the answers are equivalent,
x
1
and we arrive at a definition of negative exponents: n = x–n.
x
Example:
Simplify (by eliminating negative exponents and reducing):
a4b–1c3
a4c3c5
c8
= 7 6 1 = 3 7
a7b6c–5
a b b
a b
3 –3
17. a.
15x y
20x5y–6
b.
( 4x
−2
y 3 ) ( 2 x 4 y −5 )
2
(8 x y )
4
2 2
−1
18. a.
2 x 5 y −2
8 x3 y −5
b. (2 x 2 y 3 )3 (3 x 4 y )
c. 2−2 + 3−1
REMINDER: BASE STAYS THE SAME!
Rules for exponents:
Multiplication: Multiply the coefficients, add the exponents
ex: (3 x 5 )(2 x8 ) = 6 x13
Powers:
Take the coefficient to the power, multiply the exponents
ex: (2 x 5 )3 = 8 x15
Division:
Divide (reduce) the coefficients, subtract the exponents
ex:
20 x 7
= 4 x3
5x4
Special exponents:
Zero:
Anything except 0 to the 0 power = 1
ex: x 0 = 1; 30 = 1; (2 x − 5 y )0 = 1
Negatives:
19. Solve for x:
Negative exponents mean reciprocals
( 4 )(8 ) = 64
x +3
x +5
x +1
20. Given ∆ ABC  ∆ DEF , if AB = 36, BC = 48, AC = 60 and DE = 90, what is the length DF?
When using interval notation to describe a graph that has more than one “part,” the symbol

is used. This is the
symbol for Union. It can also be described as an “or” statement with inequalities.
Example: Describe points that are less than 4 or greater than 8
on the number line:
–5
as an inequality:
with a solution set:
using interval notation:
5
0
10
x < 4 or x > 8
{x : x < 4
or x > 8}
( − ∞, 4 )  (8, ∞ )
21. Sketch the solution on a number line and write the solution in interval notation.
b. − 4 < x < 2 or x > 9
a. x < − 2 or x ≥ 8
22. For positive real numbers a and b, define an operation
a
a•b
b = a+b
. If 3
x = 2
4
by:
Solve for x.
23. The lines y =
− 3 x + 18 , y = 0, and x = 0 enclose a triangular region. Find the area of the region.
Powers of i
Remember that i=
(
i 2 =−1
)
2
−1 . This definition is used in determining powers of i .
=
−1
i 3 = i 2 (i ) = −i
24. a. Simplify: (3 + i ) 2 + 2i − 5
b. Simplify: 2i 4 − 2i 3 + 7 − 3i 2 + 4i 7 − 10i11 + 5i14
Rationalizing
Rationalizing is used to simplify expression that have radicals that need to be eliminated. In different problems, you
may be asked to rationalize either a numerator or denominator.
To rationalize, you multiply by the conjugate if the part you are trying to rationalize is a binomial.
Example:
6
3 + 4 6 3 + 24 6 3 + 24
⋅
=
=
−13
3 − 16
3−4 3+4
25. a. Rationalize the denominator:
20
5+2
b. Rationalize the numerator:
3− 2
21
26. What is the midpoint of the segment joining points ( 5, − 2 ) and (11, 8 ) ?
 a b  10  12 
   =   , what is the ratio a : b , in simplest form?
 b −a  15  4 
27. If 
28. Solve, giving your answer in interval notation: −2 ≤
29. Solve for x:
5 − 3k
< 12
−2
a x −3a5
3x +1
7−x = a
a
30. Find k so that the line connecting the points (4, 2k ) and (3k − 2, −4) has a slope of −
4
.
3
31. The larger of two numbers is one more than twice the smaller. If the sum of the two numbers is 25, find the larger of
the two numbers.
32. What is the exact area of an equilateral triangle whose perimeter is 24?
33. Given the five numbers x, (x – 5), (x + 25), (x + 6), (x + 9). Find the value of x, if twice the average of the numbers is
equal to the median of the numbers.
3
34. Write the equation, in standard form, of the line that is parallel to y = – 2 x + 4 and passes through the point (6,–8)
35. Solve each system for all solutions (x,y):
300
180 x − 120 y =

a.  x
y 1
+ =

14 7 2
4 x +3 y = 32
b. 
2 x +3 y
= 25
125
(Hint: factor the bases)