Calculus

Calculus And Analytical

Geometry

MTH-310

Lecture 2

In this lecture we shall

Inequalities

Absolute values

Graph and coordinate

Inequality

In mathematics, an inequality is a relation that holds between

two values when they are different.

This comparison is expressed by using the inequality symbols

These four expression are equivalent

a is greater than b

a is less than b

a b

a b

The symbol means ; either or a b a b a b

These four expression are equivalent

a is greater than or equal to b a b

a is less than or equal to b a b

The symbols < and > are called strict inequality, and the symbols

and are called weak inequalities

Rules for working with inequalities symbols

Addition

Subtraction

Multiplication

Division

Summary of operation on inequalities

You can add or subtract a number on both side of an inequality.

You can multiply or divide an inequality by a positive number.

You can multiply or divide an inequality by a negative number, but you

must change the direction of inequality.

Relation between interval and inequality

Interval notation inequality notation

,

,

,

,

,

,

,

,

,

a b

a b

a b

a b

b

b

a

a

Graph

How to graph the solutions

Graph any number greater than. . .

open circle, line to the right

Graph any number less than. . .

open circle, line to the left

Graph any number greater than or equal to. . .

closed circle, line to the right

Graph any number less than or equal to. . .

closed circle, line to the left

Examples

4 35 1

2

x

Solve the inequality and draw the graph

Examples

As the altitude of a shuttle increases, an astronaut’s weight decreases

until a state of weightless is achieved. The weight of a 125-lb astronaut at

an altitude of x kilometer above see level is given by

At what altitude is the astronaut’s weight less then 5lb?

??

26400

1256400

wx

Quadratic inequalities

Quadratic function might take one of the three forms:

Usual form

Factor form

Complete square form

2

2

f x ax bx c

f x a x p x q

f x a x r s

Example

Sole the inequality

2 4 0x x

??

Example

Sole the inequality

??

1 5 0x x

Absolute values

The modulus of x , written and pronounced ‘mod x’ is defined by x

x x

x x

If

If

0,x

0.x

Example

Solve

Solution

Example

Solve

SQUARE ROOTS AND ABSOLUTE VALUES

Therefore

If a and b are real number then

Theorem

a a

ab a b

aa

b b

Graph and coordinate

A rectangular coordinate system is a pair of perpendicular coordinate lines,

called coordinate axes, which are placed so that they intersect at their origins.

The labeling of axis's with letters x and y is a common convention, but any

letters may be used. If the letter x and y are used to label the coordinate

axes, then the resulting plane is called xy-plane. In applications it is

common to use letters other than x and y is shown In the following figures,

as uv-plane and ts-plane.

For example if we take (a,b)=(4,3), then on coordinate plane

To plot a point P(a,b) means to locate the point with coordinates (a,b) in a

coordinate plane. For example, different points are plotted.

Example:

Sketch the graph of

2y x

Solve the inequality and express the solution in terms of

intervals whenever possible.

4 12 0

3

x

6 5 3x

12

2 3

x

x

Sketch the graph of the equation

2

2

2 1

4

y x

y x

Exercise