Manhattan Review Algebra

Algebra 1. Important Properties

If

and

are real numbers and

the following statements are always true?

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 9

, which of

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1. If x is a positive integer and x  3 , which of the 3y following must be true if x  y  1 ? (A) y  0 (B) y is an integer (C) y  1 (D) y  4 / 5 (E) | y | 2

2. 150012 + 150022 – 149992 – 149982 is between (A) 40,000 and 80,000 (B) 80,000 and 100,000 (C) 100,000 and 150,000 (D) 150,000 and 200,000 (E) 200,000 and 300,000

3. If k and m are numbers such that k + m = 20 and k2+ m2=289, then the value of the product km is (A) (B) (C) (D) (E)

between 20 and 60 between 60 and 100 between 100 and 150 between 150 and 200 greater than 200 11

2. Solving Quadratic Equations

This is the standard form of the quadratic equation. In the GMAT, a is usually 1. Two ways to solve: Factoring:

Quadratic formula: Remember: if D  b 2  4ac  0 , we have two distinct roots (solutions). However, if

we have one solution, and if

we have no real solutions. If solutions of opposite signs.

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, we have two

3. Absolute Values

The distance between -3 and 7 is 10, since 7 – (-3)= 10. The distance between

and

can be either

or

is the absolute value of , the distance between 0. For but for

,

If If

| x  3 | is the distance between _____ and _____.

is the distance between ____ and ______. represents the set of all real numbers that are less than ____ units from _____. 13

. and

4.

If k and p are positive constants, which of the following equations must have a negative root? x2 + kx + p=0

I.

II. x2 - kx + p=0 III. x2 + kx - p=0

I only

5. (1)

II only

III only

II & III

Is

I, II & III

?

is a positive integer

(2)

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4. Systems of Linear Equations 6.

Is

(1) (2)

7.

and

? are positive integers.

If 3x + 2y +z = 10 and x + 2y + 3z = 26, what is the value of x + y + z ?

(A) 8 (B) 9 (C) 10 (D) 11 (E) not enough information is provided 8.

Fabio bought several pairs of socks, some woolen and the others cotton. The price of the each pair of woolen socks was $3 higher than the price of each pair of cotton socks. What was the average price of the pairs of socks that Fabio bought?

(1) He bought three times as many pairs of cotton socks as woolen socks. (2) He spent twice as much money on cotton socks as he did on woolen socks.

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Orange (GMAT Review 11th Edition) PS (Problem Solving) 1,3,11,14,20,24,32,35,38,39,40 41,47,49,53,54,55,58,66,69,74,91,92,98,100,101,102,119 120,127,132,140,144,149,153,157,164,167,171,182,186 200,203,205,210,211,220,224,225,228,230,232,235,245

DS (Data Sufficiency) 3,11,12,13,21,26,29,37,38,42,44,52 54,55,56,57,60,63,67,68,70,73,75,83,88,92,96,97,98

Green (GMAT Quantitative Review) PS (Problem Solving) 1,3,7,17,22,23,26,30,38,43,49,50,51 55,62,68,72,76,78,84,96,93,94,99,104,106,107,110,111, 114,115,116,121,126,127,128,129,131,133,137,142,146, 154,155,157,160,171,175

DS (Data Sufficiency) 2,4,6,12,15,23,26,27,28,29,35,36,37 44,46,49,56,60,61,71,77,79,80,81,89,90,95,96,101,102 103,104,115,118

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