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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
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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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The aim of this document is to provide a short, self-assessment programme for students who wish to acquire a basic competence in the use of inequalities.

A number a is greater than a number b if a - b is positive. In symbols this is written as a > b.


2 > 1 because 2 - 1 = 1 is positive,

3 > - 1 because 3 - ( - 1) = 4 is positive,


- 1 > 2 is false because - 1 - 2 = - 3 is negative.

Example 1

Prove or disprove the following inequalities.


(a) As a decimal, 1/4 = 0.25 and so 0.4 - 1/4 = 0.4 - 0.25 = 0.15, which is positive. Thus 0.4 > 1/4 is true.

(b) Here (0.7) = 0.7 × 0.7 = 0.49. As a fraction 1/2 is 0.5. In this case, (0.7) - 1/2 = 0.49 - 0.5 = - 0.01, which is negative. This means that the inequality (0.7) > 1/2 is false.

For this latter example we would write (0.7) < 1/2, or in words, (0.7) is less than 1/2.

In general we say:

A number a is less than a number b if a - b is negative. In symbols this is written as a < b.

If a < b the b > a and vice versa.

Example 2

In each of the following pairs of numbers, use one of the symbols > or < to give the correct ordering of the numbers for the order in which they appear.


(a) Taking a = - 1 and b = 2 the difference a - b , becomes a - b = (-1) - 2 = - 3 , which is negative. The correct inequality is - 1 < 2.

(b) In decimal form 1/4 = 0.25 and 1/5 = 0.2. Since 0.25 - 0.2 = 0.05, and this is positive, the correct inequality is 1/4 > 1/5.

In addition to these two inequalities there are two further symbols, and . The first of these is read as greater than or equal to and the second as less than or equal to.


For each of the following pairs of numbers use one of the symbols >, <, , to give the correct ordering for the order in which they appear.



(b) (-1) = 1 and (-1/2) = 1/4 so (-1) > (-1/2).

(c) In decimal form 1/ 5 = 0.2 so 0.2 1/ 5 and 0.2 1/ 5 are both true.

(d) The solution to this can be obtained by converting the fractions to decimals as in previous cases. It may also be obtained using fractions, by writing both with the same denominator 6.


which is negative. The correct inequality is therefore


Determine which of the following inequalities is correct.

(a) 3 > 2, (b) 2 < 4, (c) 2 < 5, (d) 3 > 4


The solution to this is obtained from 3 = 9 and 2 = 8 and 9 > 8.

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