This lecture discusses inequalities, absolute values, and graphs of solutions to inequalities on a coordinate plane. Key points covered include: - Inequalities relate values that are different using symbols like < and >. Operations on inequalities follow rules like adding/subtracting a positive number or multiplying/dividing by a negative number requires changing the inequality symbol. - Absolute value expressions use | | to represent the distance from zero. Operations inside | | follow different rules than normal order of operations. - Graphing solutions to inequalities involves open/closed circles and drawing lines left/right to represent <, >, ≤, or ≥. Examples are worked through to demonstrate graphing linear and quadratic inequalities.

1. Calculus And Analytical
Geometry
MTH-310

2. Lecture 2

3. In this lecture we shall
Inequalities
Absolute values
Graph and coordinate

4. 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

5. These four expression are equivalent
a is greater than b
a is less than b
a b
a b

6. The symbol means ; either ora b a b a b
These four expression are equivalent
a is greater than or equal to ba b
a is less than or equal to ba b

7. 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

8. 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.

9. Relation between interval and inequality
Interval notation inequality notation
,
,
,
,
,
,
,
,
,
a b
a b
a b
a b
b
b
a
a
Graph

10. 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

11. Examples
4 3
5 1
2
x
Solve the inequality and draw the graph

12. 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?
??
2
6400
125
6400
w
x

13. 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

14. Example
Sole the inequality
2 4 0x x
??

15. Example
Sole the inequality
??
1 5 0x x

16. Absolute values
The modulus of x , written and pronounced ‘mod x’ is defined byx
x x
x x
If
If
0,x
0.x

17. Example
Solve
Solution

18. Example
Solve

19. SQUARE ROOTS AND ABSOLUTE VALUES
Therefore

20. If a and b are real number then
Theorem
a a
ab a b
aa
b b

21. 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.

22. 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.

24. 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.

25. Example:
Sketch the graph of
2
y x

26. Solve the inequality and express the solution in terms of
intervals whenever possible.
4 1
2 0
3
x
6 5 3x
1
2
2 3
x
x
Sketch the graph of the equation
2
2
2 1
4
y x
y x
Exercise