RI Equations and Inequalities C3 Tut (Qn)

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    [(a) 11 or 3xx  (b) No solution (c) 23 o r 3xx  ] 2 Without using a calculator, solve the following inequalities. (a) 5 11 x x   (b) 31 2 x xx   [4] [(a) 13 x (b) 2 or 0 xx  ] 3 Without using a calculator, solve the following inequalities. (i) 23 7x (ii) 26 1 x [(i) 25 x (ii) 11 or 26xx  ] 4 By sketching appropriate graphs on a single di agram, find the exact solution set to the following inequalities. (i) 33 1xx  (ii) 11 1xx [(i)  ,1 (ii) 11,, 10 12           ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________________ Tutorial C3: Equations and Inequalities Page 2 of 4 5 A cinema sells tickets at three different pric es, 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 co s A 9 6 4 $162.03 B 7 5 3 $128.3 6 C 10 4 5 $158.5 0 Write down and solve the equations to find the cost of a ticket for each of the age categories. [4] [x =7.65, y=9.85, z =8.52] Section B (Discussion Questions) 1 Without using a calculator, solve the following inequalities. (a)  2 29x (b)   221 0xx x   (c) 224 5 0xx  (d) 2 2 2 1(2 ) xx xx   (e) 2 2 3411 23 xx xx    [(a) 5 or 1x x  , (b) 2x , (c)  , (d) 1 or 2 3xx   , (e) 71 or 152 x x   ] 2 Without using a calculator, solve the inequality 2 2 1 02 xx xx   . [4] [ 21 x ] 3 Without using a calculator, solve the inequality 2 2 29 19 xx x    . Hence solve the inequalities (i) 2 2 29 19 xx x    , (ii)   2 2 2 ln ln 9 1 ln 9 xx x    . [ 30 o r 3xx   , (i) 0o r 3o r 3xx x  , (ii) 33e1 o r exx   ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________________ Tutorial C3: Equations and Inequalities Page 3 of 4 4 (i) By using an algebraic method, solve the inequality 35 41 2 x x x   . [4] (ii) Hence, solve the inequality 2 22 35 41 2 x x x   . [4] [(i) 14 2x , (ii) 11 22 x  ] 5 Sketch the graph of 225 3yx x  for 24 x , indicating the axial intercepts. Hence write down the range of values of k for which the equation 225 3x xk  has 4 distinct roots. [ 490 8k ] 6 Find the set of values of x such that 23 2 2 .xx Solve this using two different methods: (a) algebraic method, (b) graphical method. [  : 0.4 or 6xx x   ] 7 The solution set to the inequality 23 4xa x  is 9 ,5 b . Find a and b . [ 17 9a , b = 9] 8 (i) On the same axes, sketch the graphs of yx a and 1ya x where 01 ,a and hence solve the inequality 1.xa a x  [4] (ii) Deduce the solution to the inequality 1 1.aaxx [2] [(i) 11 x  , (ii) 1 or 1x x  ] 9 (i) The first three terms of a sequence are given by 1 10u  , 2 6u  , 3 5u  . Given that nu is a quadratic polynomial in n, find nu in terms of n. [4] (ii) Find the set of values of n for which nu is greater than 100. [2] [un = 1.5n2 – 8.5n + 17, {n  ℤ+ : n  11}]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ________________________________ Tutorial C3: Equations and Inequalities Page 4 of 4 10 A curve C has equation 2 ayb x cx , where a, b and c are constants. It is given that C passes through the points with coordinates (1.6, 2.4) and ( 0.7, 3.6) , and that gradient of C is 2 at the point where 1x . (i) Find the values of a, b and c, giving your answers correct to 3 decimal places. [4] (ii) Find the x-coordinate of the point where C crosses the x-axis, giving your answer correct to 3 decimal places. [2] [(i) 3.593, 5.187, 7.303abc   , (ii) 0.589 ] 11 A movie theatre supports a charity event by providing discounted movie tickets at the following prices based on age group. Age group Discounted ticket price 12 years $6 13 to 49 years $9 50 years (senior citizen) $4 The event organiser plans to spend $700 to sponsor 100 participants of which at least 35 of them are senior citizens. Find the possible number of participants for each age group. [4] [ 50 58 or 60 37 40 xx yy zz      ]