1. MTH108 Business Math I
Lecture 2

2. Basic Concepts (Equations)
Review
Basic Concepts (Equations)
algebraic expression
variable
equation
solution, solution set
types of equations
polynomial
equivalent equations

3. Review Solving First Degree Equation in One Variable
Solving Second Degree Equation in One Variable
by factorization
by quadratic formula
real, distinct roots
one real root
no real root

4. Section 3
Inequalities
Interval
Solving Inequalities

5. 1.3.1 Inequalities
1.3.1 Definition Inequalities express the condition that two quantities are not equal. Inequalities are denoted by the symbols < and >.
1.3.2 Examples
3 < is less than 5
x > x is greater than 100
0 < y < y is greater than
zero and less than 50

6. 1.3.3 Definition Strict inequalities are those statements in which items are never equal to one another.
They are of three types.
Absolute inequality which
are always true.
e.g. 3 < 5.
Conditional inequality true under certain conditions e.g. x > 100.
Double inequality
involving two comparisons.
e.g. 0 < y < 50.

7. Use: to compare quantities.
This comparison can be represented on the real line.
Interpretation If a and b R, then there are three possibilities.
a = b a b
a < b a b
a > b b a

8. 1.3.4 Definition Statements in which one item can be equal to the other item are called non-strict inequalities. They are denoted by the symbols ≤ and ≥ respectively.

9. y ≤ x y is less than or equal to x. a ≤ x <b x is greater than or
1.3.5 Examples
x+3 ≥ x+3 is greater than
or equal to 15.
y ≤ x y is less than or equal
to x.
a ≤ x <b x is greater than or
equal to a and x is
less than b.
a b
[ x )

10. 1.3.2 Interval Notation
With the help of example 1.3.5(iii), we now define the interval notations and their meanings Definition An interval is the set of numbers that lies between two numbers a and b. They can be specified using the following notation: (a, b) = {x | a < x < b}

11. The notation (a, b) represents the open interval with endpoint values a and b not included in the interval.
A closed interval [a, b] includes the end points a and b. More precisely,
[a, b]={x| a ≤ x ≤ b}
Half-open intervals include one endpoint but not the other.
(a, b] = {x| a < x ≤ b }
[a, b) = {x| a ≤ x < b }

12. Graphical representation of intervals
(a, b) a b ( x ) [a, b] a b [ x ] (a, b] a b ( x ] [a, b) a b [ x )

13. 1.3.7 Exercise: Sketch the following intervals.
(-2, 1)
[1, 3]
[-3, -1)

14. 1.3. 3 Solving Inequalities Similar to solving equations
We attempt to determine the set values satisfying the inequality
Isolate the variable on one side of the inequality using algebraic operations

15. 1.3.8 Rules for Inequalities
Rule 1 If the same number is added or subtracted from both sides of an inequality, the resulting inequality has the same sense as the original inequality. Symbolically,
If a < b, then a+c < b+c and
a-c < b-c
Rule 2 If both sides of an inequality are multiplied or divided by the same positive number, the

16. if a < b and c > 0, then ac < bc and a/c < b/c
resulting inequality has the same sense as the original one. Symbolically,
if a < b and c > 0, then ac < bc and a/c < b/c
Rule 3 If both sides of an inequality are multiplied or divided by the same negative number, then the resulting inequality has the reverse sense of the original one. Symbolically,

17. if a < b and c < 0, then ac > bc and a/c > b/c
Rule 4 Any side of an inequality can be replaced by an expression equal to it. Symbolically,
if a < b and a = c, then c < b
Rule 5 If the sides of the inequality are both positive or both negative, then their reciprocals are unequal in the reverse sense, e.g.
2 < 4 but ½ > 1/4

18. If 0 < a < b and n > 0, then
Rule 6 If both sides of an inequality are positive and raised to the same positive power, then the resulting inequality has the same sense as the original one. Symbolically,
If 0 < a < b and n > 0, then
and

19. 1.3.9 Examples 3x + 10 ≤ 5x – 4 3x + 10 +4 ≤ 5x -4 +4 rule 1
3x+14 ≤ 5x simplify
3x-3x +14 ≤ 5x-3x rule 1
14 ≤ 2x simplify
14/2 ≤ 2x/ rule 2
7 ≤ x simplify
Values of x which are greater than or equal to 7 satisfy the inequality, i.e. [7, ∞).(conditional inequality)

20. 6x -10 ≥ 6x+4
6x ≥ 6x rule 1
6x ≥ 6x simplify
6x-6x ≥6x-6x rule 1
0 ≥ simplify
False inequality
4x+6 ≥ 4x-3
4x+9 ≥ 4x rule1, simplify
9 ≥ rule1, simplify
absolute inequality

21. -2x+1 ≤ x ≤ 6-x
-2x+1 ≤ x left ineq. first
1 ≤ 3x rule 1, simplify
1/3 ≤ x rule 2
x ≤ 6-x solving right ineq.
2x ≤ rule1, simplify
2x/2 ≤ rule 2
x ≤ simplify
1/3 ≤ x ≤ combining both ineq.
The interval satisfying this double inequality is [1/3, 3].

22. 1.3.10 Second Degree Inequalities
If an inequality involves a higher- order algebraic expression, we can solve the inequality by factorising it. For example, the simplest is to understand the second degree inequality. The solution of the inequality or the interval where the inequality holds will be determined by considering different cases.

23. x-3 = 0 and x-3 < 0 x-2 = 0 and x-2 < 0
Examples 1)
Soln. (x-3)(x-2) ≤ factorising
Now we have two possibilities;
(x-3) ≤ 0 and (x-2) ≤ 0
First condition: x-3 ≤ 0 gives
x-3 = 0 and x-3 < 0
x= and x < 3
Second condition: x-2≤ 0 gives
x-2 = 0 and x-2 < 0
x= and x < 2

24. First condition: x-3 < 0 further gives x-2 > 0 and x-3 < 0
Second condition: x-2 <0 further gives
x-3 > 0 and x-2 < 0
x > 3 and x < 2
Interval of solution [2, 3]

25. 2)
Soln. (x-5)(x+3)> factorising
First condition: x-5>0 and x+3>0
which gives x>5 and x>-3
Second condition: x-5<0 and x-3<0
which gives x<5 and x<-3
Interval of solution: