Transcript Algebra Released Questions 1-18

Standard 1.1
1. Is the equation 3(2 x – 4) = −18
equivalent to 6 x − 12 = −18?
a) Yes, the equations are equivalent by the
Associative Property of Multiplication.
b) Yes, the equations are equivalent by the
Commutative Property of Multiplication.
c) Yes, the equations are equivalent by the
Distributive Property of Multiplication
over Addition.
d) No, the equations are not equivalent.
Answer
• First, always look to see if the solution is
the correct answer. In this case it is. If it
was not, then “d” would have been the
answer.
• After seeing that the equation is correct, you
look at what is different between the two
equations. When you do this you would
notice you distributed the “3”.
• ∴ the answer is “c”
Standard 2.0
2.
a)
b)
c)
d)
16  8
3
4
6
9
10
Answer
•
16  8
3
• When a number is inside the radical sign,
it’s asking for the square root of the
number.
• When a number is inside of a radical sign
with a cubed number located on the left,
it’s asking for the cubed (2 • 2 • 2) of the
number
• ∴ 4 + 2 = 6. The answer is “b”
Standard 2.0
3. Which expression is equivalent to x 6 x 2
a)
b)
c)
d)
x4x3
x5x3
x7x3
x9x3
Answer
• When looking at x 6 x 2 , all you need to
remember is how do you multiply variables
with exponents (you add the exponents). In
this case x 6 x 2 = x 8, ∴ you are looking for
the same answer.
a)x 4 x 3 = x 7
b)x 5 x 3 = x 8
c)x 7 x 3 = x 10
d)x 9 x 3 = x 12
• ∴ “b” is your answer.
Standard 2.0
4. Which number does not have a
reciprocal?
a) -1
b) 0
1
c) 100
d) 3
Standard 2.0
• Reciprocal is a number that can be times
another to equal 1.
• Several problems on the test were meant to
see if you can follow directions
• In this problem the key word was “not”.
In this case you need to remember 0 times
anything equals 0. This is called the
Zero-Product Property.
• ∴ the answer can only be “b”.
Standard 3.0
5. What is the solution for this equation?
|2x – 3| = 5
a)
b)
c)
d)
x = - 4 or x = 4
x = - 4 or x = 3
x = - 1 or x = 4
x = - 1 or x = 3
Standard 3.0
• First thing to address is to know the
equation |2x – 3| = 5 has an absolute value
in it. This means you will have two
answers and need to set up two equations.
2x – 3 = - 5
2x – 3 + 3 = - 5 + 3
½ • 2x = - 2 • ½
x =-1
∴ the answer is “c”
2x – 3 = 5
2x – 3 + 3 = 5 + 3
½ • 2x = 8 • ½
x =4
Standard 3.0
6. What is the solution set of the inequality
5 - |x + 4| ≤ - 3
a)
b)
c)
d)
-2≤x≤6
x ≤ - 2 or x ≥ 6
- 12 ≤ x ≤ 4
x ≤-12 or x ≥ 4
• Absolute value is again in this problem but
it also has a negative sign in front of it and
an inequality sign (≤) in the problem.
5 - |x + 4| ≤ - 3
5 – (x+4) ≤ - 3
5 –x – 4 ≤ - 3
- x + 1 ≤ -3
- x + 1 - 1 ≤ -3 - 1
-x≤-4
x≥4
5 - |x + 4| ≤ - 3
5 + (x+4) ≤ - 3
5+x+4≤ -3
x+9≤-3
x + 9 - 9 ≤ -3 - 9
x ≤ - 12
• ∴ the answer is “d”.
• Less than /greater than or equal to.
Standard 4.0
7. What is the equation to
5x – 2(7x + 1) = 14x
a)
b)
c)
d)
- 9x – 2 = 14x
- 9 x + 1 = 14x
- 9 x + 2 = 14x
12x – 1 = 14x
• The key to answer this problem is
knowing the Distributive Property of
Multiplication over Addition and how to
add like terms.
• 5x – 2(7x + 1) = 14x
•
5x –2(7x +1) = 14x
•
5x – 14x -2 = 14x
•
-9x – 2 = 14x
• ∴ the answer is “a”.
•Dist Prop
•Add Like Terms
Standard 4.0
8. What is the equation to
4(2 – 5x) = 6 – 3(1 – 3x)
a) 8x = 5
b) 8x = 17
c) 29x = 5
d) 29x = 17
• Distributive Property of Multiplication
over Addition and like terms.
• 4(2 – 5x) = 6 – 3(1 – 3x)
• 4(2 – 5x) = 6 – 3(1 – 3x) •Dist Prop
•Add Like Terms
•
8 – 20x = 6 – 3 + 9x
• 8 – 8 – 20x – 9x = - 8 + 3 + 9x – 9x
•
- 29x = -5
•“x” can not be
•
29x = 5
negative
• ∴ the answer is “c”.
Standard 5.0
9. The total cost (c) in dollars of renting a
sailboat for n days is given by the
equation
c = 120 + 60n
If the total cost was $360, for how many
days was the sailboat rented?
a) 2
b) 4
c) 6
d) 8
• The problem gives you the equation and
the total cost ($360), all you need to do is
to take your time. If “c” is the total cost,
and they tell you the total cost is $360
then solve the equation
360 = 120 + 60n
360 - 120 = 120 – 120 + 60n
1
/60 • 240 = 60n • 1 /60
4=n
∴ the answer is “b”
Standard 5.0
10. Solve:
3(x+5) = 2x + 35
Step 1:
3x+15 = 2x + 35
Step 2:
5x + 15 = 35
Step 3:
5x = 20
Step 4:
x=4
• Which is the first incorrect step in the
solution above
a) Step 1
b) Step 2
c) Step 3
d) Step 4
10. Solve:
3(x+5) = 2x + 35
Step 1:
3x+15 = 2x + 35
Step 2:
5x + 15 = 35
Step 3:
5x = 20
Step 4:
x=4
• Incorrect is the key word. The two
important items to remember are getting
the variable you are solving for alone and
add like terms. In this case, 3x was
added to 2x, the error was it was suppose
to be plus -2x. End result of the error is
the incorrect step. ∴ “b” is the answer
Standard 5.0
11. A 120-foot-long rope is cut into 3 pieces.
The first piece of rope is twice as long as
the second piece of rope. The third piece
of rope is three times as long as the
second piece of rope. What is the length
of the longest piece of rope?
a) 20 Feet
b) 40 Feet
c) 60 Feet
d) 80 Feet
• A 120-foot-long rope is cut into 3 pieces.
•
•
•
•
•
1st Piece = twice as long as the second
3rd Piece = three times as long as the 2nd
1st piece = 2x
3rd piece = 3x
2nd piece = x
x + 2x + 3x = 120
6x = 120
x = 20
• It’s a trick, what is the longest piece?
• x = 20
2(20) = 40
3(20) = 60
• ∴ “c” is the answer, 60 feet.
Standard 5.0
12. The cost to rent a construction crane is
$750 per day plus $250 per hour of use.
What is the maximum number of hours
the crane can be used each day if the
rental cost is not to exceed $2500 per
day?
a)
b)
c)
d)
2.5
3.7
7.0
13.0
• 750 + 250 x ≤ 2500 is your equation
• 1st , apply what you know, 250 • 10 = 2500
• 2nd , rule out 13 for it’s too high
• 3rd, look at a number close to 10 then add
750. If it does not exceed 2500, that is your
answer. Try x = 7.
750 + 250 (7) ≤ 2500
750 + 1750 ≤ 2500
2500 ≤ 2500
• Finally, know the sign. If there is a line
under the inequality sign, then the
numbers can be equal. ∴ “c” is your
answer.
Standard 24.1
13.Which number serves as a
counterexample to the statement below?
All positive integers are divisible by 2 or 3.
a)
b)
c)
d)
100
57
30
25
Standard 24.1
• Vocabulary is important in this question.
Knowing that “counterexample” means
what number proves the statement wrong.
• Knowing your tables are important, if you
don’t divide (the word “divisible” was
used) each number by 2 or 3.
• Remember if it is even, 2 will go into it so
don’t waste your time. If it is odd, try 3.
• 100 and 30 are divisible by 2.
• 57 and 30 are divisible by 3.
• This leaves 25 ∴ the answer is “d”.
Standard 24.2
14. What is the conclusion of the statement
in the box below?
If x 2 = 4, then x = -2 or x = 2.
a)
b)
c)
d)
x2=4
x = -2
x=2
x = -2 or x = 2
Standard 24.2
• Vocab, vocab, vocab. In this problem
YOU USE YOUR LOGIC.
• Knowing that conclusion means what can
be assumed in the problem all you need to
do is match what is after “then” to the
answers below.
• The part “If x 2 = 4,” is your basic
question, so it rules out “a”
• Both “b” and “c” are only partial
answers because the part “then x = -2 or
x = 2.” tells you they both work. ∴ “d” is
your answer.
Standard 24.3
15.The chart below shows an expression
evaluated for four different values of x.
X
1
2
6
7
x2 + x + 5
7
11
47
61
Josiah concluded that for all positive values
of x, x 2 + x + 5 produces a prime number.
Which value of x serves as a counterexample
to prove Josiah’s conclusion false.
a) 5
b) 11
c) 16
d) 21
Standard 24.3
• Vocabulary and logic are key.
• 1st, “prime number” means it can’t be
divided by an integer (excluding 1 and
itself) without leaving a remainder.
(example 3, 5, 11)
• 2nd, use what you know. Without
replacing x with the numbers to see if it
works, you know that 5 times anything
will be divisible by 5. With a product of 5
times 5, with two more 5’s added, the
number will be divisible by 5. ∴ the only
logical answer is “a”.
Standard 25.1
16. John’s solution to an equation is shown
below. (Which property for Step 2)
Given:
x 2 + 5x + 6 = 0
Step 1: (x + 2) (x + 3) = 0
Step 2: x+2 = 0 or x + 3 = 0
Step 3:
x = -2 or
x = -3
a) Multiplication property of equality
b) Zero product property of multiplication
c) Commutative property of multiplication
d) Distributive property of multiplication
over addition.
• 1st, eliminate those you know (c and d).
Comm Property (3 • 4 = 12) is a simple
multiplication problem and you have not
distributed anything eliminate them.
• 2nd, knowing that the Multiplicative
Property of Equality only means to
multiply each side of an equation or
both expressions, by the same number
(a • c = b • c) or ( 2 • 5 = 4 • 5)
• You are down to one answer, “b”. If
you want to make sure, look at its name
Zero Product, or a zero answer to
multiplication problem.
17. Stan’s solution (25.2)
Given:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
a)
b)
c)
d)
n + 8(n + 20) = 110
n + 8n + 20 = 110
9n + 20 = 110
9n = 110 – 20
9n = 90
9n / = 90 /
9
9
n = 10
Stan’s solution is correct.
Stan made a mistake in Step 1
Stan made a mistake in Step 3
Stan made a mistake in Step 5
• 1st, use what you know. The problem is
correct if Steps 1, 3, and 5 are correct
which would make “a” your answer.
• 2nd, remember that each step is the result
of the previous line. For example Step 1 is
the result of Dist Prop 8 (n+20) in the
“Given” line.
• When looking at each step you should
notice that Step 1 is incorrect. 8 (n + 20)
would be “8n +160” not “8n + 20”
• ∴ the answer is “b”.
Standard 25.3
18. When is this statement true?
The opposite of a number is less than the original
number.
a) The statement is never true.
b) This statement is always true.
c) This statement is true for positive
numbers.
d) This statement is true for negative
numbers.
•
•
•
•
•
1st, remember what you have to answer, “when
is the statement true”
2nd, if you can answer “c” or “d”, you will
eliminate the first two.
Next, replace x with numbers (1, & -1).
x–x<x
1–1<1
or
-1 – (-1) < -1
Determine which one is true. If you notice,
the second one is not true. - (-1) = + 1
If you notice 1 – 1, or the opposite of a positive
number is less than the original number 1.
• ∴ “c” , your answer, is true and this eliminates
both “a” and “b”