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Miscellaneous Equations
Operations with Fractions
Undefined Rational Expressions
Writing Equations for Lines Using Sequences
Intersections of Lines and Conics
Graphing Linear Equations
Solving Equations with Log Terms and Other Terms
Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
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Values of Symbols for Which Fractions are Undefined

Here we simply remind you that several properties of fractions of the following special forms continue to apply to fractions in which the numerator and/or denominator are algebraic expressions. We have already discussed and illustrated these properties with reference to numerical fractions. They are

  as long as b is not also equal to zero.
  is undefined (or infinite) as long as b is not also equal to zero.
  is undefined (or indeterminate)

The words “undefined” or “indeterminate” here do not mean that people are presently not clever enough to figure out what these expressions must mean. Instead, they indicate that such expressions are numerical nonsense – it is impossible to interpret them in a way that makes numerical sense, or that assigns to them an actual meaningful numerical value.

From this list, we see that essentially, whenever the denominator of a fraction evaluates to zero, we may speak of the value of that fraction as being undefined.

The slight twist with algebraic fractions is that if the denominator is an algebraic expression, then the value of the denominator depends on the values assigned to any symbols that are present. For some values of those symbols, the denominator may evaluate to a nonzero value and so no problem arises. However, there may be some values of the symbols for which the denominator does evaluate to zero, and so the fraction as a whole becomes undefined.


Example 1:

For what values of x is the fraction undefined?


Fractions are undefined when the denominator has a value of zero. Thus, this fraction will be undefined for any value of x that makes

4x – 7 = 0

But this is just a very simple equation. Solving, we get

4x = 7

and so


4(1.75) – 7 = 7 – 7 = 0.

Thus, the given fraction is undefined only when x = 7 / 4 = 1.75


Example 2:

For what values of x is the fraction undefined?


Again, we need only look at the denominator. The fraction as a whole will be undefined whenever the denominator is equal to zero, that is, whenever

3x 2 – 48 = 0

This is an equation for x which is not too hard to solve:

3x 2 – 48 = 0


3x 2 = 48

and so

So any value of x that can be squared to give 16 will satisfy this equation. But 16 = 4 2 and 16 = (-4) 2 , so there would appear to be two values of x that satisfy this equation and hence make the denominator of the original equation equal to zero:

So, the original fraction is undefined if x = +4 and if x = -4. (You can check by direct substitution that both of these values of x result in the original fraction having the form b / 0, where b 0.)

A final reminder: fractions are undefined in the way we’ve been discussing here only when the denominator evaluates to zero. If the numerator evaluates to zero, but the denominator is nonzero, the fraction itself just has the value of zero and no problem results. Thus, in Example 1 above, if we substitute x = -3 / 5, we get:

This poses no mathematical problem. “Undefinedness” of the sorts referenced in items (ii) and (iii) of the list at the beginning of this note only arises when the denominator evaluates to zero.

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