TMJC H2 Chapter 4 Equations & Inequalities Learning Package 2024

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Chapter 4 Equations and Inequalities TMJC 2024 Page 1 of 26 H2 Mathematics (9758) Chapter 4 Equations and Inequalities Core Concept Notes Success Criteria: SURFACE DEEP TRANSFER Solving an equation  Solve polynomial equations by factorisation and/or quadratic formula  Solve polynomial equations using graphing calculator  Use of graphing calculator to solve a non-polynomial equation by sketching graphs System of linear equations  Solve the system of linear equations using the graphing calculator Inequalities  Represent inequalities on a real number line (e.g. 3x ) using open/closed circle  Interpret the mathematical language “and” and “or” on a real number line  Solve linear or quadratic inequalities by hand or graphical methods  Solve simple inequalities involving modulus by hand or graphical methods  Use of graphing calculator to solve inequalities (graphical methods) System of linear equations  Interpret the solutions found using a graphing calculator Inequalities  Solve inequalities of the form f ( ) 0g( ) x x  (< , ≥ , ≤ ) by hand or graphical methods, where f ( )x and g( )x are linear or quadratic expressions.  Solve inequalities of the form a < f ( )xb (≤ ) by hand or graphical methods System of linear equations  Formulate a contextual problem into possible mathematical equations and provide possible solutions, and interpret/reflect the solutions in the real world context Inequalities  Formulate a contextual problem into possible mathematical inequalities and provide possible solutions, and interpret/reflect the solutions in the real world context (Examples can be in subsequent topics such as Arithemtic and Geometric Series, and Binomial / Normal Distributions.)

Chapter 4 Equations and Inequalities TMJC 2024 Page 2 of 26 §1 Finding the Numerical Solution of an Equation Recap: The solution (roots/zeros) to an equation is/are the value(s) which satisfies (satisfy) the equation. To find the solution (roots/zeros) of an equation, we can use one of the following methods : (a) Algebraic Factorisation/ Formula (b) Using Application: PlySmlt2 on a Graphing Calculator (GC) (c) Sketching graphs using a Graphing Calculator (GC) (a) Algebraic Factorisation/Formula (When a question needs an EXACT solution) Example 1 Find the values of x satisfying the equation 2 20xx+ − = . Algebraic Factorisation 2 20xx+ − = ( )( )2 1 0 2 or 1 xx xx + − =  =− = Formula 2 20xx+ − = ( )( ) ( ) 21 1 4 1 2 21 19 2 2 or 1 x xx −  − −= −= =− = (b) Using Application: PlySmlt2 on a Graphing Calculator (GC) This application can only be used to solve polynomials up to degree 10. Example 2 Solve the equation 32 2 5 6x x x− − =− . 32 2 5 6x x x− − =− 32 2 5 6 0x x x− − + = Using GC, 2 or 1 or 3x x x=− = = Using PlySmlt2 (POLY ROOT FINDER) to find roots of an equation with ONE variable This APPS can be used only if the equation to be solved is a polynomial equation i.e. 2 2 1 0 0n na x a x a x a+ + + + = . Bring everything to one side Factorise & solve

Chapter 4 Equations and Inequalities TMJC 2024 Page 3 of 26 Step 1: (i) Press A and select 4:PlySmlt2. (ii) Press e to enter the main menu. (iii) Select 1: POLYNOMIAL ROOT FINDER Step 2: (i) Select the required parameters for the equation 32 2 5 6 0x x x− − + = . (Using Example 2 on pg. 2) (ii) Press % to go NEXT. Step 3: (i) Key in the coefficients of the equation 32 2 5 6 0x x x− − + = . (ii) Press % to go SOLVE.

Chapter 4 Equations and Inequalities TMJC 2024 Page 4 of 26 (c) Sketching graphs using a Graphing Calculator (GC) Example 3 Solve the equation 2 e2xx −+= . Note: The e quation involves a non-polynomial expression i.e. e x− . Hence, PlySmlt2 application cannot be used. Method (a): Using “intersection” method Step 1: Sketch the LHS and RHS of the given equation as two graphs on the same diagram. Step 2: Find the points of intersections between these two graphs. Step 3: The x-coordinates are the roots to the given equation. Interpretation: By sketching the LHS and RHS as two graphs 2 1 e xyx −=+ and 2 2y = , solving the equation 2 e2xx −+= means finding values of x where 12yy= (that is, the x- values of the points of intersection)  the roots of the equation 2 e2xx −+= are −0.537 and 1.32 (to 3 s.f.). This method can be used to find the solution for any type of equation with ONE variable. The “intersection” method (Finding the intersection points of two graphs) Step 1: (i) Press !. (ii) Enter the equation 2 e xyx −=+ in Y1 (iii) Press e. (iv) Enter the equation 2y= in Y2 (v) Press %. (Press # followed by 2: Zoom In to change the zoom settings) y x O 1.32 –0.537

Chapter 4 Equations and Inequalities TMJC 2024 Page 5 of 26 Step 2: Press `$ and select 5: intersect (Keystrokes: `$ opens the CALCULATE MENU for graphs) Step 3: Press e once. The calculator will prompt you: First curve? Check that it is the selected equation is Y1 Press e again. The calculator will prompt you: Second curve? Check that it is the selected equation is Y2 Press e again. The calculator will prompt you: Guess? Using < and >, move the blinking cursor nearer to the intersection that you want to find. Press e one last time and t he x-value and y-value of the intersection will be shown. (Repeat the process to find the other intersection)

Chapter 4 Equations and Inequalities TMJC 2024 Page 6 of 26 Method (b): Using “zeros” method Step 1: Rewrite the given equation such that the RHS is now equals to zero. i.e. 22 e 2 e 2 0xxxx −−+ =  + − = Step 2: Sketch the LHS of the equation as one single graph. Step 3: Find the x-intercepts of the graph. The x-intercepts are the roots to the equation 2 e 2 0xx −+ − = . Interpretation: Sketching the LHS as one graph 2 1 e2xyx −= + − and finding the x-intercept is equivalent to drawing two graphs 2 1 e2xyx −= + − and 2 0y = and hence solving the equation 2 e 2 0xx −+ − = means finding values of x where 12yy= (that is, the x- values of the x-intercepts). From the GC, the graph of 2 e2xyx −= + − is  the roots of the equation 2 e2xx −+= are −0.537 and 1.32 (to 3 s.f.). The “zeros” method (Finding the x-intercepts of a graph) Step 1: (i) Press !. (ii) Enter the equation 2 2xy x e −= + − in Y1 Press e. (iii) Press %. (Press # followed by 2: Zoom In to change the zoom settings) y x O 1.32 –0.537

Chapter 4 Equations and Inequalities TMJC 2024 Page 7 of 26 Step 2: Press `$ and select 2: zero (Keystrokes: `$ opens the CALCULATE MENU for graphs) Step 3: Press e once. The calculator will prompt you: Left Bound? Using < and >, move the blinking cursor to the left of the x-intercept that you want to find. Press e again. The calculator will prompt you: Right Bound? Using < and >, move the blinking cursor to the right of the x-intercept that you want to find. Press e again. The calculator will prompt you: Guess? Do not move the cursor! Press e one last time and t he value of the x-intercept will be shown . (Check that the y-value is equal to zero) (Repeat the process to find the other two intersections)

Chapter 4 Equations and Inequalities TMJC 2024 Page 8 of 26 §2 Syst