CHAPTER SUMMARY

I. Simplifying Rules

Simplifying Rules

Solving an equation is simplifying to get a single variable equal to a constant, such as x = 4.
When solving an equation, you need to do the same thing to both sides. View equations as a balancing act. Add, subtract, divide, multiply, square, or take the root of both sides of an equation.

Adding Polynomials

To add or subtract polynomials, add like terms. You can use a horizontal or vertical format.

Cross Multiplication

Cross multiplication can be used to solve rational equations. Rational equations contain fractions with variable expressions in the numerator or denominator.

Distributive Property

The distributive property is used to simplify expressions.

a(b + c) = (a × b) + (a × c) = ab + ac

Multiplying Polynomials

  • To multiply a monomial by a monomial, multiply the numerical coefficients and each variable.
  • To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial.
  • To multiply two polynomials, multiply each term of the first expression by each term of the second. Then combine like terms.

Solving Equations with Two Variables

There are two basic methods: Substitution and Addition. With Substitution, solve for one variable then substitute it in the other equation. You can also solve equations with more than one variable by adding (or subtracting) the equations.

Three or More Variables

system of linear equations consists of two or more linear equations with the same variables. Solving systems of equations with more than two variables is generally the same as solving for systems with two variables.  There are just more steps because there are more variables.

One incorrect assumption is you can solve a linear system only if you have the same number of equations as variables. Another false assumption is that a linear system always has one answer. Systems can have one, none, or an infinite number of answers.

II. Quadratic Equations

Quadratic Equations

The most common polynomials on the GMAT are binomials and quadratics. Multiplying two binomials gives a quadratic.

Multiplying Binomials

A quick way to multiply two binomials is to use FOIL (First, Outer, Inner, Last), then combine like terms.

There are three common patterns in multiplying binomials. Memorizing these patterns will help in multiplying and factoring binomials and polynomials:

  • (a + b)2 = a2 + 2ab + b2
  • (a – b)2 = a2 – 2ab + b2
  • (a + b)(a – b) = a2 – b2

Quadratic Equations in Standard From

quadratic equation is an equation of degree 2 that can be written in standard form
ax2 + bx + c = 0 where a ≠ 0. However, quadratic equations are rarely in standard form. To solve, you may have to rewrite the equations.

Factoring Quadratic Equations

Factoring is the simplest way to solve most quadratic equations.

How to factor x2 + bx + c = 0:

  1. If b and c are positive, then the two factors of c are positive and their sum is b.
  2. If b is negative and c is positive, then the factors of c are negative and their sum is b.
  3. If c is negative, then the factors of c have different signs and their sum is b.

The Quadratic Formula

The solutions of the quadratic equation ax2 + bx + c = 0, a ≠ 0, are:

x = \dfrac{-\textit{b} \pm \sqrt{\textit{b}^{\displaystyle{2}} \,-\, 4\textit{ac}}}{2\textit{a}}

Solving Radical Equations

radical equation is an equation with terms that are the square root of a variable. To solve a radical equation, isolate the radical on one side, then square both sides.

III. Complex Expressions

Complex Expressions

To divide polynomials, you can use the same principles you use when multiplying them.

  • Two methods for dividing a polynomial by a monomial are factoring and dividing each term.
  • Two methods for dividing a polynomial by a binomial are factoring and long division.

Simplifying Exponential Expressions

Looking at the answers before starting to solve the problem is a good strategy for questions with exponents.  You may match the answer format by just simplifying or estimating, rather than calculating to get a value.

Complex Fractions

complex fraction is a fraction that has a fraction in the numerator or denominator. In other words, it is a fraction divided by a fraction. Complex fractions can contain variable expressions. To simplify, multiply by the reciprocal, then factor and cancel any common factors.

Adding Polynomial Fractions

When adding (or subtracting) algebraic fractions, follow the same process as adding number fractions.

  1. Find the least common denominator (LCD) of the fractions.
  2. Write equivalent fractions using the LCD.
  3. Add (or subtract) the numerators.
  4. Simplify and reduce the resulting fraction.

How Many Solutions?

The number of solutions for polynomial equations can vary.You need to be careful when performing operations on an equation that you don’t lose a possible solution or get an extraneous solution. You might lose a solution if you divide both sides by the variable, and you might get an extraneous solution if you square both sides of an equation.

Polynomials and Radicals

You can apply the properties of multiplying and dividing polynomials to simplify equations with radicals. Remember that the simplest form of an expression does not have radicals in the denominator.

Variables as Exponents

Apply the properties of polynomials and exponents when exponents are variables. Rewrite terms to have the same base. Write an equation from the exponents.

IV. Inequalities

Inequalities

An inequality compares two quantities or expressions.  There are four inequality symbols:  <,  ≤,  >, and  ≥. A compound inequality is two separate inequalities that show a range of values.

Absolute Value

Absolute value is another way to express a range of values. The absolute value equation | x | = 2 means the distance between x and 0 is 2.  There are two solutions, 2 and -2.

The absolute value inequality | x | < 2 means the distance between x and 0 is less than 2, so x > -2 and x < 2.

The inequality | x | ≥ 2 means the distance between x and 0 is greater than or equal to 2, so x ≤ -2 or x ≥ 2.

Solving Inequalities

Solving an inequality uses many of the same rules as solving an equation.

  1. The same number or algebraic expression may be added or subtracted from both sides of an inequality.
  2. The same positive number may multiply or divide both sides of an inequality.

There is one rule for solving inequalities that is different than the rule for solving equations:

  • If a negative number multiplies or divides both sides of an inequality, the inequality sign must be “reversed.”

Note:  You can only divide or multiply by a variable or expression if you know the sign (positive or negative) of the variable or expression.

Solving Systems of Inequalities

Inequalities can be added if both inequality symbols point the same way. You can multiply one of the inequalities by -1 so the symbols match. Always add, not subtract, inequalities.

Inequalities in Triangles

There is one inequality in triangles that is often used on the GMAT.

  • The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

V. Working With Formulas

Working With Formulas

Formulas are just equations with specific formats. Formula questions on the GMAT will supply values for some variables and expressions and ask you to find the other pieces in the formula. Be sure to familiarize yourself with formulas that commonly appear on the GMAT.

Substitute Given Values

You need to be able to substitute values into a formula, making sure to put the values into the right places.

Rewriting Formulas

You can solve an equation for any of the variables. Rewriting a formula by solving for one of the variables uses the same steps as solving an equation.

Multiple Solutions

Solving a formula for one of the variables is a good approach when you are asked to find multiple values for that variable. You can also substitute numbers to find values and ranges of values for a variable.

Using More Than One Formula

Using two formulas that have the same variables is similar to solving systems of equations.
When making comparisons, don’t immediately solve all the way. Keeping the variables or constants that are in both formulas can make for easier comparisons.

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