RVHS H2 Math Inequalities Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Inequalities 1 Inequalities 1. CJC/2013/1/Q4 (a) Using an algebraic method, solve the inequality 2 4 1.32 x xx + +− [5] (b) Hence solve the inequality 2 42 4 1.23 x xx − +− [2] 2. ACJC/2016/I/1 Without the use of a calculator, solve the inequality 2 25 03 w w − − . [2] Hence solve ( ) 2 2 5 sin 03 yx y − − , given that 3 2x   . [3] 3. DHS/2016/I/3 (i) Without using a calculator, find the exact solution of the inequality 44. 2x x− + [3] (ii) Hence solve solution of 45 1x x− + . [2] 4. TJC/I/2 Without the use of a calculator, solve the inequality 2 3 26 11 x xx −+ . [5] Hence, find the range of values of x which satisfies 2 3 14 12 11 x xx e ee −+ . [2] 5. ACJC/2010/II/3 Find in terms of a, the range of values of x that satisfy the inequality ln 2 0 ax x  − , where 1a  . [4] 6. VJC Prelim/2020/01/Q2 Solve the inequality 2 2 23 0,( 1) 1 xx ax a x ++ − + + for 𝑎 ∈ ℝ,  𝑎 ≠ 0,  𝑎 < 1. You need to distinguish the cases where a is positive and a is negative. [5]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Inequalities 2 7. AJC/2011/1/3 Solve the inequality ( )( ) ( )( ) 2 0x a x b x x c +−  − , where 𝑎, 𝑏, 𝑐 ∈ ℝ+ and 0 c b a   . Hence, find the range of values of x which satisfy ( )( ) ( )( ) 2 ln ln 0 ln ln x a x b x c x +−  − . [5] 8. HCI/2013/Prelim/P1/Q1 Given that a is a positive real number, solve the inequality ( ) 2 2 2 0 1 xx x a x a −+  + − − , leaving your answer in terms of a . [3] 9. DHS/2013/I/Q2 Without the use of a calculator, solve the inequality ( )1 1 1 .xxx +  + [3] Hence, solve 21 e e e 0x x x −− + −  . [2] 10. MI/2021/I/6 (i) Sketch the curve with equation 1y ax= − , where a is a positive constant. State, in terms of a, the equations of the asymptotes and coordinates of any intersections with the x-axis and y-axis. On the same diagram, sketch the line with equation ( )y b x a=− , where b is a positive constant. [4] (ii) Find, in terms of a and b, the root of the equation ( )1 b x aax =−− . [3] (iii) Hence solve the inequality ( )1 b x aax −− . [2] 11. MI Prelim/2020/01/Q1 Sketch the graph of ( ) 2ln 4yx=− , stating the equations of any asymptotes. [2] By adding a suitable curve on your graph above, solve the inequality [3] ( ) 22ln 4 8 0xx − + − 

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Inequalities 3 12. TJC Prelim/2022/01/Q3 (i) On the same axes, sketch the graphs of y x a=+ and 1y ax=+ where 0 1,a and hence solve the inequality 1.x a ax+  + [4] (ii) Deduce the solution to the inequality 1 1.aaxx+  + [2] 13. YIJC Prelim 9758/2023/01/Q3 (a) Find the exact roots of the equation 23 8 3 3 .x x x+ − = − [3] (b) On the same axes, sketch the curves with equations 23 8 3y x x= + − and 3.yx=− Hence solve exactly the inequality 23 8 3 3 .x x x+ −  − [4] 14. MI Prelim 9758/2024/02/Q1 Without the use of a calculator, solve the inequality ( ) ( ) 2 57 33 xx xxx +−  −− . [4] Hence solve ( ) ( ) 2 57 33 xx xxx −+  ++ . [2] 15. EJC Prelim 9758/2024/02/Q1 It is given that a is a real number such that 1a − . (a) Factorise 3 2 2 3x ax a x a− + − . [1] (b) Hence solve the inequality 3 2 2 3 0( )( 1) x ax a x a x a x − + − ++ . [4] (c) Using your answer to part (b), solve the inequality 3 22 4 8 0( 2)( 1) x x x xx + + + −+ . [3] 16. NYJC Prelim 9758/2025/P2/Q1 (a) Find the set of values of x for which 3 123 xx − − . [2] (b) Without using a calculator, solve 2 18 15 14 x xx + +−  − . [4] 17. SAJC Prelim 9758/2025/P2/Q1 (i) Solve the inequality 23 4 2xx −  + . [4] (ii) Hence, solve 2 2 3 1 214 xxxx +− . [2]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Inequalities 4 Answers 1. (a) 1 or 3xx −  ; (b) 1 or 1xx −  2. 33 w−   or 5 2w  ; 33 y−   or 5 2y  or 5 2y − 3. (i) 2 or 1 1xx      − −   + (ii) 2 2 x    − −   + 4. 112 2x ; 110 ln 4x 5. 1 1 8 1 1 8 044 aa x or x− + + +   6. 11 x a 1 or 1.xx a 7. xa− or 0 xb and xc ; 0 axe − or 1 bxe and cxe 8. 1 xa−   9. 0 or 1xx ; 0x  10. (ii) 1xa b =+ , (iii) xa or 1a x a b   + 11. Asymptotes: x = 2− , x = 2 2.63 2 or 2 2.63xx−   −   12. (i) −1 < x < 1 (ii) 𝑥 < −1 or 𝑥 > 1 13. (a) 3 17 2 − or 0 or 7 3− (b) 3 17 7 3 17 or 0 or 2 3 2x x x− − − + −    14. 70 or , 3 5x x x   Hence: 70 or , 3 5x x x  −  − 15. (a) ( )( ) 22xa a x+ − (b) 1 ora x x a  −  − (c) 2x  or 2x − 16. (a) {𝑥 ∈ ℝ: 2 ≤ 𝑥 ≤ 8}. (b) 7 or 3 5 3 5 or 2x x x − − −   − +  17 (i) 57 or 233 xx  = − (ii) 3 3 1 or7 5 2xx  = −