MTH108 Business Math I Lecture 2

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

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

Section 3 Inequalities Interval Solving Inequalities

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 < 5 3 is less than 5 x > 100 x is greater than 100 0 < y < 50 y is greater than zero and less than 50

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.

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

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.

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

1.3.2 Interval Notation With the help of example 1.3.5(iii), we now define the interval notations and their meanings. 1.3.6 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}

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 }

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

1.3.7 Exercise: Sketch the following intervals. (-2, 1) -2 -1 0 1 [1, 3] 1 2 3 [-3, -1) -3 -2 -1

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

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

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,

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

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

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/2 rule 2 7 ≤ x simplify Values of x which are greater than or equal to 7 satisfy the inequality, i.e. [7, ∞).(conditional inequality)

6x -10 ≥ 6x+4 6x -10+10≥ 6x+4+10 rule 1 6x ≥ 6x+14 simplify 6x-6x ≥6x-6x+14 rule 1 0 ≥ 14 simplify False inequality 4x+6 ≥ 4x-3 4x+9 ≥ 4x rule1, simplify 9 ≥ 0 rule1, simplify absolute inequality

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

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.

x-3 = 0 and x-3 < 0 x-2 = 0 and x-2 < 0 Examples 1) Soln. (x-3)(x-2) ≤ 0 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=3 and x < 3 Second condition: x-2≤ 0 gives x-2 = 0 and x-2 < 0 x=2 and x < 2

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]

2) Soln. (x-5)(x+3)>0 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: