TMJC H2 Chapter 4 Equations & Inequalities Discussion Solutions 2024

Chapter 4 Equations and Inequalities TMJC 2024 Page 1 of 22 H2 Mathematics (9758) Chapter 4 Equations and Inequalities Discussion Solutions Level 1 Questions 1 Solve the equation 221 11 2 0x x   algebraically. Q1    221 11 2 0 7 1 3 2 0 x x x x       1 2 or7 3x x  3 methods to solve quadratic equations algebraically: 1) Factorisation (shown in solution) 2) Complete the square (try on your own) 3) Quadratic Formula (try on your won)

Chapter 4 Equations and Inequalities TMJC 2024 Page 2 of 22 2 Using a graphing calculator, solve the following equations (a) 3 22 23 41 115 0x x x    , (b) 1 2 2ln 0 xx   . Q2 (a) 3 22 23 41 115 0x x x    Using GC,      1.46 3 s.f. or 4.87 3 s.f. or 8.08 3 s.f.x x x   (b) 1 2 2ln 0 xx   Using GC, x = 0.187 or x = 2.16 y x O 2.16 0.187 x = 0 Solving equations using a GC: - For polynomial equations (up to degree 10) we can use the PolyRootFinder in the APP. - We can use the graphical method (sketching one or two graphs) to determine the roots (x-value of the intersection points).

Chapter 4 Equations and Inequalities TMJC 2024 Page 3 of 22 3 Solve the system of linear equations: 31 2 4 1 2 3 4 1 2 3 4 1 2 3 4 10 4 2 0 xx x x x x x x x x x x x x x x                 Q3 31 2 4 1 2 3 4 1 2 3 4 1 2 3 4 10 4 2 0 xx x x x x x x x x x x x x x x                 Using GC, .x1 = 4, x2 = 3, x3 = 2, x4 = 1.

Chapter 4 Equations and Inequalities TMJC 2024 Page 4 of 22 4 By drawing number lines, simplify the following ranges of values of x: (a) 3 8 x   and 4x , (b) 3 8 x  or 2 5 x   . Q4 (a) 3 8 x   and 4x 4 8 x   (b) 3 8 x  or 2 5 x   2 8 x   5 Without the use of a calculator, solve the following inequalities (a) 2 5 4x x  , (b) 29 6 8 0x x   , (c) 2 7x  , (d) 2 1 3x  . Q5 (a) 2 5 4 2 5 5 2 x x x x       (b)    29 6 8 0 3 2 3 4 0 x x x x       2 4 or3 3x x  (c) 2 7x  7 2 7 9 5 x x        (d) 2 1 3x  2 1 3 or 2 1 3 2 2 or 2 4 1 or 2 x x x x x x         3 4 x 8 2 3 x 5 8 “and” means intersection of the two regions. “or” means union of the two regions.

Chapter 4 Equations and Inequalities TMJC 2024 Page 5 of 22 6 By sketching appropriate graphs, solve the following inequalities (a) 1 ln xx  , (b) 2 4 1 1 0x x    . Q6 (a)  1For ln , 1.76 3 s.f.x xx   (b)   2For 4 1 1 0, 0 0.586 or 3.41 4 3 s.f.x x x x        x y ݔ= 0 ݕ= 0 ݕ= 1 ݔ ݕ= lnݔ 1.76, 0.567) y x Method 1: Sketching of two graphs (both LHS and RHS) to solve the inequality. Method 2: Sketching of one graph to solve the inequality.

Chapter 4 Equations and Inequalities TMJC 2024 Page 6 of 22 Level 2 Questions 7 A VCR manufacturing company produces three models: model