ACJC 2025 Equations & Inequalities Summary

Anglo-Chinese Junior College 2025 H2 Mathematics 9758: Equations & Inequalities / Summary / Page 1 of 2 SUMMARY: Equations and Inequalities Graphs of Polynomials Make sure the coefficient of the highest power is positive so that graph always start from the top right-hand corner. Factors (x - a)n - n is odd: cuts the x-axis - n is even: Turning point at x-axis E.G. (x +1)2(2 - x)(3+ x)3 £ 0 23( 1) ( 2)(3 ) 0x x x + − +  2, 1or 3x x x  =− − Other Methods - number line - substitute values using GC Rational Functions ▪ Do not cross multiply! Make one side zero. ▪ Always factorise! If there are no roots, that means the expression is always positive or negative. Use discriminant or complete square to justify. ▪ For exact roots of quadratic equations, apply the formula if necessary, i.e. 2 4 2 −  −b b ac a . ▪ Do not include the x-values that makes the denominator 0. E.G. 12 13 xx +−  bring all the terms to one side make one side zero 12 ( 1) 03− + − xx 12 ( 1)( 3) 03 xx x − + − −   2 2 2 multiply by ( 3) make coefficient of highest power ve 12 2 3 03 2 15 03 ( 5)( 3)( 3) 0, 3 x xx x xx x x x x x − + − + + − −− −  − + −   3 or 3 < 5xx −  E.G. x2 + 2x + 5 x2 + 3x - 28 < 0 x2 + 2x + 5 = (x +1)2 + 4 > 0 for all x  2 2 2 5 can be divided without affecting sign 1 03 28 ++ +− xx xx 1 0 7 4( 7)( 4) xxx  −  +− Modulus Function ,0 ,0  = −  Two very useful inequalities x < a Û -a < x < a x > a Û x < -a or x > a E.G. 2x +1 < x - 2 ( 2) 2 1 2 2 2 1 AND 2 1 2 1 33 x x x x x x x xx − −  +  − − +  + +  −   − no intersection, therefore no solution. 2 1− 3−

Anglo-Chinese Junior College 2025 H2 Mathematics 9758: Equations & Inequalities / Summary / Page 2 of 2 Squaring both sides can only be used when both sides are always positive. Bring to one side and factorise after squaring both sides! E.G. 2x + 3 < 3x +1 Square both sides ( ) ( )  22 22 2 2 factorise using ( )( ) 4 5 2 3 3 1 (3 1) (2 3) 0 (3 1 2 3)(3 1 2 3) 0 (5 4)( 2) 0 2 or a b a b a b xx xx x x x x xx xx − = + − +  +  + − +   + + + + − −   + −    − Solve Inequalities using GC E.G. Find the values of x for which xex is within 0.5 of x + x2 + 1 2 x3 . Find x s.t. 23 1 2e ( ) 0.5xx x x x− + +  Sketch 23 1 2e ( )xy x x x x= − + + and y = 0.5 . \-1.75 < x <1.535 Equations Polynomial Equations Solve using GC app PlySmlt2 Solving Equations Using GC E.G. Find the points of intersections of the curves ln( 2)yx=− and 2yx=+ Plot ln( 2) 2y x x= − − + and find the intersection with the x-axis using “zero” System of Linear Equations Form linear equations and solve using GC app PlySmlt2 ▪ Equations with more unknowns than equations E.G. At a funfair, Royston spent $76 on 20 food items consisting of cup corn, nachos and churros to share with his friends. The cost of each portion of cup corn, nachos and churros are $3, $4 and $5 respectively. Given that he bought at least 5 portions of each food item, how many portions of each food item did he buy? Solution: Let the number of portions of cup corn, nachos and churros be x, y and z respectively. 3 4 5 76 20 x y z x y z + + = + + = From GC, x = 4 + z 5 1 z   y = 16 – 2z 5 5.5 z   z = z 5 Since x, y and z must be at least 5 , we deduce that z = 5, x = 9, y = 6