Example: 2x+7 = 0

x = -7/2

Algebra can farther be divided into the following:

**1. Simplifying Algebraic Expressions**

Example:

3x + 6ax^{2} + 15xa^{2} |

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

x = (y-b)/m

3. Exponents

The power

*n*is called the exponent and

*a*is called the base in

*a*. For example in 7

^{n}^{4}, 4 is the exponent and 7 is the base.

The rules or laws of exponents are:

MUST-KNOW:

** Rule 1 : a^{n} **×

**×**

Rule 2 :

*a*=^{m}*a*^{n }^{+ m}Rule 2 :

*a*^{n}

Rule 3 : (

*b*= (^{n}*a b*)^{n}Rule 3 : (

*a*)^{n}^{m}=*a*^{nm} Rule 1 : 2^{2} × 2^{5} = 4 × 32 = 128 and 2^{2} × 2^{5} = 2^{7} = 128

Rule 2 : 2^{3} × 4^{3} = 8 × 64 = 512 and 2^{3} × 4^{3} = 8^{3} = 512

Rule 3 : (3^{2})^{3} = 9^{3} = 729 and (3^{2})^{3} = 3^{6} = 729

**MUST-KNOW :**

*a*= 1/^{−n}*a*^{n}

Example:

2^{−4} = 1/2^{4} = 1/16 and (¼)^{−1} = 1/(¼) = 4

You would think that a question like

*x*

^{2}= 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 |

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 | ||

- 1) = 1 -(x | 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.

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

**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 - (2 - x) = 2 - (x - 2)

2 - 2 + x = 2 - x + 2

x = 4 - x

2x = 4

x = 2

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!!!

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