RI Equations and Inequalities C3 Tut Sect A (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ____________________________________ Tutorial C3: Equations and Inequalities Page 1 of 4 Tutorial 3: Equations and Inequalities Section A (Basic Questions) 1 Without using a calculator, solve the following inequalities. (a) 234 1 0xx  (b) 222 1 0xx  (c)   22 934 9xx x x    Solution: (a) 234 1 0xx  2 2 44 4 ( 3 ) ( 1 ) 44 1For 3 4 1 0, or 1 2(3) 6 3xx x          For 234 1 0xx  , 11 or 3xx  OR: 234 1 0xx  (3 1)( 1) 0 11 or 3 xx xx     (b) 222 1 0xx  2 2 1120 22 11 1But 2 0 22 2 So there is no real solution for the inequality. x xx          (c)   22 934 9xx x x               2 2 2 2 2 33 3 4 9 34 93 0 35 6 0 32 0 Since 3 0 and 3, 2 0 2 Solution is 2 3 or 3 xx xx x xx x x xx x xx xx x xx xx                        1 x

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________________ Tutorial C3: Equations and Inequalities Page 2 of 4 2 Without using a calculator, solve the following inequalities. (a) 5 11 x x   (b) 31 2 x x x   [4] Solution: (a) 5 1, 11 x xx      5 101 26 01 26 1 0 13 x x x x xx x        (b) 31 ,2 , 02 x xxx x             2 2 2 2 2 31 02 (3 ) ( 2) 0(2 ) 22 0(2 ) 22 20 22 20 11 2 0 Since 1 1 0 , 2 0 2 or 0 x xx xx x xx xx xx xxx x xxx x xx x xx xx xx                           

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________________ Tutorial C3: Equations and Inequalities Page 3 of 4 3 Without using a calculator, solve the following inequalities. (i) 23 7x (ii) 26 1 x Solution: (i) 23 7x 72 37 42 1 0 25 x x x           (ii) 26 1 x 26 1 o r 26 1 11 or 62 x x xx          OR: Squaring both sides, 22 22 (2 6 ) 1 (2 6 ) 1 0 (6 3)(6 1) 0 x x xx      11 or 26xx  4 By sketching appropriate graphs on a single diagra m, find the exact solution set to the following inequalities. (i) 33 1xx  (ii) 11 1xx Solution: (i) 33 1xx  At point A, 33 1xx  1x For 33 1xx  ,  ,1x  Note: the method of squaring both sides can only be used when both sides of the inequalities are positive. This method is also longer than using the identity: f( ) f( ) o r f( ) ,xa x a x a    where 0.a

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________________ Tutorial C3: Equations and Inequalities Page 4 of 4 (ii) 11 1xx At A,  111 1 10xx x    At B,  111 1 12xx x  Solution set is 11,, 10 12           5 9740/2012/01/Q1 A cinema sells tickets at three different prices , depending on the age of the customer. The age categories are under 16 years, between 16 and 65 years, and over 65 years. Three groups of people, A, B and C, go to the cinema on the same day. The numbers in each age category for each group, together with the total cost of the tickets for each group, are given in the following table. Group Under 16 years Between 16 and 65 years Over 65 years Total cost A 9 6 4 $162.03 B 7 5 3 $128.36 C 10 4 5 $158.50 Write down and solve the equations to find the cost of a ticket for each of the age categories. [4] Solution: Let $x, $y and $z be the cost of a ticket for age categories “Under 16 years”, “Between 16 and 65 years” and “over 65 years” respectively. Then, 9 6 4 162.03 7 5 3 128.36 10 4 5 158.50 xyz xyz xyz    By GC, we have x =7.65, y=9.85, z =8.52.