Standard 1.11. 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 2.02.

a)4

b)6

c)9

d)10

3 816

Answer•

• 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”

3 816

Standard 2.03. Which expression is equivalent to x 6 x 2

a)x 4 x 3

b)x 5 x 3

c)x 7 x 3

d)x 9 x 3

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 2.04. Which number does not have a

reciprocal?

a)-1

b) 0

c)

d) 3

100

1

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 3.05. What is the solution for this equation?

|2x – 3| = 5

a)x = - 4 or x = 4

b)x = - 4 or x = 3

c)x = - 1 or x = 4

d)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.

∴ the answer is “c”

2x – 3 = - 52x – 3 + 3 = - 5 + 3 ½ • 2x = - 2 • ½

x = - 1

2x – 3 = 52x – 3 + 3 = 5 + 3 ½ • 2x = 8 • ½

x = 4

Standard 3.06. What is the solution set of the inequality

5 - |x + 4| ≤ - 3a) - 2 ≤ x ≤ 6b) x ≤ - 2 or x ≥ 6c) - 12 ≤ x ≤ 4d) 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.

• ∴ the answer is “d”. • Less than /greater than or equal to.

5 - |x + 4| ≤ - 35 – (x+4) ≤ - 3 5 –x – 4 ≤ - 3 - x + 1 ≤ -3

- x + 1 - 1 ≤ -3 - 1

- x ≤ - 4 x ≥ 4

5 - |x + 4| ≤ - 35 + (x+4) ≤ - 3 5 + x + 4 ≤ - 3 x + 9 ≤ - 3

x + 9 - 9 ≤ -3 - 9

x ≤ - 12

Standard 4.07. What is the equation to

5x – 2(7x + 1) = 14x

a) - 9x – 2 = 14x

b) - 9 x + 1 = 14xc) - 9 x + 2 = 14xd) 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 4.08. What is the equation to

4(2 – 5x) = 6 – 3(1 – 3x)

a) 8x = 5

b) 8x = 17c) 29x = 5d) 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)

• 8 – 20x = 6 – 3 + 9x

• 8 – 8 – 20x – 9x = - 8 + 3 + 9x – 9x

• - 29x = -5

• 29x = 5 • ∴ the answer is “c”.

•Dist Prop•Add Like Terms

•“x” can not be negative

Standard 5.09. 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) 4c) 6d) 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 + 60n1 /60 • 240 = 60n • 1 /60

4 = n

∴ the answer is “b”

Standard 5.010. 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 1b) Step 2c) 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 5.011. 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 Feetb) 40 Feetc) 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 5.012. 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) 2.5b) 3.7c) 7.0

d) 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 24.113.Which number serves as a

counterexample to the statement below?

All positive integers are divisible by 2 or 3.

a) 100b) 57c) 30

d) 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 24.214. What is the conclusion of the statement

in the box below?

If x 2 = 4, then x = -2 or x = 2.

a) x 2 = 4b) x = -2c) x = 2

d) 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 24.315.The chart below shows an expression

evaluated for four different values of x.

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

X x 2 + x + 5

1 7

2 11

6 47

7 61

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 25.116. 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: n + 8(n + 20) = 110

Step 1: n + 8n + 20 = 110

Step 2: 9n + 20 = 110

Step 3: 9n = 110 – 20

Step 4: 9n = 90

Step 5: 9n /9 = 90 / 9

Step 6: n = 10a) Stan’s solution is correct.

b) Stan made a mistake in Step 1

c) Stan made a mistake in Step 3

d) 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 25.318. 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”