TOPIC 2 EQUATIONS, INEQUALITIES AND ABSOLUTE VALUES Subtopic 2.2 : Inequalities
Learning Outcomes: At the end of this lesson, students should be able to: (a) relate the properties of inequalities. (b) Find the linear inequalities. (c) Find the quadratic inequalities by algebraic or graphical approach.
Properties of inequalities Example 1. If a < b and c is any number, then a + c < b + c 2 < 4 and c = 3 , Then 2+ ( 3) < 4 +(3) 5 < 7 2. If a < b and c is any number, then a – c < b – c 2 < 4 and c = 3 , Then 2 – (3) < 4 – 3 - 1 < 1
Properties of inequalities Example 3. If a < b and c > 0, then a.c < b.c - 4 < - 3 and 5 > 0 , then - 4 ( 5 ) < - 3 ( 5 ) - 20 < - 15 4. If a < b, and c < 0, then a.c > b.c -4 < - 3 and – 2 < 0 , Then -4(-2) > -3(-2) 8 > 6
Properties of inequalities Example 5. If and , 0, 1 1 then a b a b a b 2 4, 1 1 then 2 4
The sign of inequality must be changed when it is multiplied by a negative number Tip
The following inequality signs are used to form Linear inequalities . > is greater than ≥ is greater than or equal to < is less than ≤ is less than or equal to Symbols Description
Definition of Linear Inequalities Linear inequalities are inequalities that can be written as: ax + b > 0 , ax + b ≥ 0 , ax + b < 0 or ax + b≤ 0 with a and b are real numbers and a ≠ 0. http://portal.kmpp.mat rik.edu.my/mod/resour ce/view.php?id=2387
(a) 11 – 2 3 2 x x Find the values of x which satisfy the inequality EXAMPLE 1 7 3 (b) 10 4 x
Solution set : : 1 x x Interval form : [1 , ) ( ) 11 – 2 3 2 a x x 11 – 2 3 6 x x 11 – 2 3 6 x x 5 5 x x 1 http://portal.kmpp.mat rik.edu.my/mod/resour ce/view.php?id=2392
7 3 (b) 10 4 x 7 3 40 x 3 33 x x 11 Solution set : : 11 x x Interval form : , 11 http://portal.kmpp.matri k.edu.my/mod/resource/ view.php?id=2395
Learning Tip • The double inequalities in the form a < b < c can be written as a < b and b < c . • The solution of a < b must be combined with the solution of b < c by the intersection between them (due to ‘and’)
Find the set values of x which satisfy the inequalities. a 10 5 3 2 x b 20 3 8 4 x x EXAMPLE 2
a 10 5 3 2 x 5 3 10 and 5 3 2 x x 3 15 x 3 3 x x 5 x 1 1 5 Solution set = { :1 5} x x SOLUTION
( ) 20 3 8 4 b x x SOLUTION : x x 20 3 8 and x3 8 4 2x 12 x 6 3x 4 8 3x 12 x 4 Solution set= {x :6 x 4} -6 4 http://portal.kmpp.mat rik.edu.my/mod/resour ce/view.php?id=2399
Definition Quadratic inequalities can be written as : with a , b and c are real numbers and a ≠ 0. 2 ax bx c 0 2 ax bx c 0 2 ax bx c 0 2 ax bx c 0
Quadratic inequalities can be solved using i. Graphical methods ii. Algebraic method - Real number line -Table of signs http://portal.kmpp.mat rik.edu.my/mod/resour ce/view.php?id=2402
Find the set of values that satisfy (i) Graphical method 2 x x 5 4 0 1 4 solution set: {x: x<1 or x>4} Interval form : (-∞,1) (4,∞) ( 4)( 1) 0 x x EXAMPLE 3 2 x x 5 4 0 http://portal.kmpp.mat rik.edu.my/mod/resour ce/view.php?id=2403
1 4 x2 – 5x + 4 > 0 (x-1)(x-4) >0 Let x-1>0 , x-4>0 x >1 x >4 + + + - - - (+) (-) (+) Solution set: {x: x<1 or x>4} Interval form : (-∞,1) (4,∞) (ii) Algebraic method (Real Number line) http://portal.kmpp.matri k.edu.my/mod/resource /view.php?id=2404
+ - + x - - + -4 x -1 - + + x x < 1 1 < x < 4 x > 4 (ii) Algebraic method (Table of sign) Solution set : x x or x : 1 4 Interval form : (-∞,1) (4,∞) x x 1 4 http://portal.kmpp.mat rik.edu.my/mod/resour ce/view.php?id=2406
Solve this equation -x2 -7x + 8 0 2 x x7 8 0 2 x x 7 8 0 Multiply -1 at both sides ( 8)( 1) 0 x x Solution: 8 1 Solution set = x x : 8 1 Interval form = 8,1 EXAMPLE 4 http://portal.kmpp.matr ik.edu.my/mod/resourc e/view.php?id=2407