GMAT Math - Algebra

Algebra on GMAT Problem Solving

This type of problem is based on making equations and solving for a variable.

Algebraic expression is a mathematical expression that contains a variable is called an algebraic expression.

Example: 2x+7 = 0
x = -7/2

Algebra can farther be divided into the following:

1. Simplifying Algebraic Expressions

Example:

3x + 6ax2 + 15xa2

3x

Here you would have to reduce this to a simpler form to solve. We can see that 3x is a common factor in both denominator and numerator and thus solve it to 1+2ax_5a. This one is easy but you might see something tougher on GMAT.

2. Equations: Linear Equations & Solving Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. Linear equations can have one, two, three or more variables. A common form of a linear equation in the two variables x and y is

Solving this type of equation has been explained above with an example. You have to take x on one side of the = and the rest of the expression to the other side as follows :

x = (y-b)/m

3. Exponents

The power n is called the exponent and a is called the base in aⁿ. For example in 7⁴, 4 is the exponent and 7 is the base.

The rules or laws of exponents are:

MUST-KNOW:

Rule 1 : aⁿ × a^m = a^{n + m}
Rule 2 : aⁿ × bⁿ = (a b)ⁿ
Rule 3 : (aⁿ)^m = a^nm

Examples:

Rule 1 : 2² × 2⁵ = 4 × 32 = 128 and 2² × 2⁵ = 2⁷ = 128
Rule 2 : 2³ × 4³ = 8 × 64 = 512 and 2³ × 4³ = 8³ = 512
Rule 3 : (3²)³ = 9³ = 729 and (3²)³ = 3⁶ = 729

MUST-KNOW : a⁻ⁿ = 1/aⁿ

Example:

2⁻⁴ = 1/2⁴ = 1/16 and (¼)⁻¹ = 1/(¼) = 4

"Dual" Answer problem for exponents
You would think that a question like x² = 25 would be simple, but it is not because there are two answers: -5 and +5. Any number raised to an even numbered exponent will always be positive. The reason for this is that -5 × -5 is 25.

	The "no answer" trick x2 + 25 = 0 This is a "trick" statement because there isn't an answer to it. This breaks down into x2 = -25. x2 can't be a negative number. Therefore, x2 + 25 = 0 can't have a real answer.

4. Inequalities

An inequality is simply a comparison of two quantities or expressions.

a	a is less than b
a < b	a is less than or equal to b
a > b	a is greater than b
a > b	a is greater than or equal to b

Example:
Solve for x:

4 - x/2 ≤ 2

-x/2 ≤ -2

-x ≤ -4

x ≥ 4

x can be any number greater than or equal to 4.

5. Absolute Value

The absolute value of a number, | |, is always positive. In other words, the absolute value symbol eliminates negative signs.

For example, | -7 | = 7. Caution, the absolute value symbol acts only on what is inside the symbol, | |. For example, -| -7 | = -(+7) = -7. Here, only the negative sign inside the absolute value symbol is eliminated.

Example:

If a, b, and c are consecutive integers and a <>

I. b - c = 1
II. abc/3 is an integer.
III. a + b + c is even.

(A) I only (B) II only (C) III only (D) I and II only (E) II and III only

Let x, x + 1, x + 2 stand for the consecutive integers a, b, and c, in that order. Plugging this into Statement I yields b - c = (x + 1) -(x + 2) = -1. Hence, Statement I is false.

As to Statement II, since a, b, and c are three consecutive integers, one of them must be divisible by 3. Hence, abc/3 is an integer, and Statement II is true.

As to Statement III, suppose a is even, b is odd, and c is even. Then a + b is odd since even + odd = odd. Hence, a + b + c = (a + b) + c = (odd) + even = odd. Thus, Statement III is not necessarily true. The answer is (B).

Another example for dual answer problem

6 - 5|x - 1| = 1
-5|x - 1| = -5	Subtract 6 from both sides.
|x - 1| = 1	Divide by -5 from both sides. Once you have isolated the absolute value, get rid of absolute value sign by creating two scenarios (one negative and one positive).
Set to negative		Set to positive
-(x - 1) = 1	Negative scenario	(x - 1) = +1	Add 1
-x + 1 = 1	Minus 1 from both sides	x = 2
x = 0

Here x can be 0 as well as 2.

6. Functions

Defined functions are very common on the GMAT, and most students struggle with them. Yet once you get used to them, defined functions can be some of the easiest problems on the test. In this type of problem, you will be given a symbol and a property that defines the symbol.

Example:

Define x # y by the equation x # y = xy - y. Then 2 # 3 =

(A) 1 (B) 3 (C) 12 (D) 15 (E) 18

From the above definition, we know that x # y = xy - y. So all we have to do is replace x with 2 and y with 3 in the definition: 2 # 3 = 2(3) - 3 = 3. Hence, the answer is (B).

Example:

Define the symbol * by the following equation: x* = 2 - x, for all non-negative x. If (2 - x)* = (x - 2)*, then x =
(A) 0
(B) 1
(C) 2
(D) 4
(E) 6

(2 - x)* = (x - 2)*
2 - (2 - x) = 2 - (x - 2)
2 - 2 + x = 2 - x + 2
x = 4 - x
2x = 4
x = 2 The answer is (C).

Well, so much for Algebra on the GMAT quantitative section. The key again is practice practice and practice. The rules are simple and so is the math. The only thing you are being tested on is your presence of mind and understanding of the concept! Happy practicing!!!