RI Equations and Inequalities C3 Lect Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _______________________________ Chapter 3: Equations and Inequalities Page 1 of 20 Chapter 3: Equations and Inequalities SYLLABUS INCLUDES • Formulating an equation, a system of linear equations, or inequalities from a problem situation • Solving an equation exactly or approximately using a graphing calculator • Solving a system of linear equations using a graphing calculator • Solving inequalities of the form f( ) 0g( ) x x > where f( x) and g( x) are linear expressions or quadratic expressions • Concept of |x |, and use of relations xa b ab x ab−<⇔−<<+ and or xa b x ab x ab−>⇔<− >+ • Solving inequalities by graphical methods PRE-REQUISITES 1. Sketching of simple graphs (e.g.: straight lines, quadratic, cubic, quartic, exponential and logarithmic) 2. Factorisation of quadratic and cubic expressions 3. Simultaneous equations 4. Completing the square for quadratic expressions 5. Applying the quadratic formula CONTENT 1 Inequalities 1.1 Properties of inequalities 1.2 Solving inequalities by factorisation, completing the square, applying the quadratic formula and sketching simple graphs 1.3 Solving inequalities involving rational functions by reducing the inequalities to one involving polynomials 1.4 Solving inequalities involving modulus 2 System of Linear Equations

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________________________ _______________________________ Chapter 3: Equations and Inequalities Page 2 of 20 INTRODUCTION Many problems in real life can be formulated by means of inequalities. They are inevitably an important part of mathematics – you will find yourself solving inequalities in many topics in mathematics. In this chapter, we shall look at some basic concepts of inequalities and the various techniques of solving them. 1 Inequalities Inequalities are mathematical expressions involving the symbols >, <, ≥, ≤ and ≠. In solving an inequality like f(x) > 0, where f(x) is an expression in x, we have to find all values for x such that when substituted into f(x), satisfies the given inequality. For example, to solve 2 1x ≥ , we may rewrite the inequality as 2 10x −≥ and proceed to factorise the LHS to get ( )( )1 10xx− +≥ . Thereafter, we have 1 or 1xx≤− ≥ as a solution to 2 10x −≥ . We write 2 1 0 1 or 1x xx− ≥ ⇔ ≤− ≥ , where the symbol ⇔ is used to indicate that the statements are equivalent. Note: “ 2 1 1xx≥ ⇒ ≥± ” is incorrect! We CANNOT solv e inequalities in the same way as we solve equations. 1.1 Properties of inequalities The following basic concepts are useful in solving inequalities. 1. a > b and c > 0 ⇒ ac > bc. 2. a > b and c < 0 ⇒ ac < bc. 3. a b > 0 ⇔ ab > 0. 4. a b < 0 ⇔ ab < 0. Note: • The inequality sign remains unchanged if a positive constant is multiplied or divided to both sides of an inequality. • The inequality sign is reversed if a negative constant is multiplied or divided to both sides of an inequality.

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________________________ _______________________________ Chapter 3: Equations and Inequalities Page 3 of 20 Example 1 Test yourself: Given a and b are non-zero real constants, state, with a reason, whether the following statements are true or false. (a) If 0,a> 5 > 4aa True. Since 54> , multiplying by a which is positive does not change the sign. (b) If 0,b< 8 <10bb False. Since 8 10< , multiplying by b which is negative changes the sign. (c) 3 0 3 > 0 aa >⇒ True. Multiplying by a2 which is positive does not change the sign. (d) 9 0 9 >0a abb <⇒ Fa lse . Multiplying by b2 which is positive does not change the sign. (e) 2 1 2 bb <⇒ < False. Let 1.b=− 2So we have 1 but 2 1.1− < >− (f) 1100ab ab>> ⇒ << T rue . It can be easily proven by considering the graph of 1y x= 1.2 Solving inequalities by factorisation, completing the square, applying the quadratic formula and sketching simple graphs Factorisation, completing the square and applying the quadratic formula are important techniques that can help us to solve inequalities. Knowing how to sketch simple graphs, like straight lines, graphs involving quadratic, cubic or quartic expressions in x , and exponential and logarithmic graphs can also help us to solve the inequalities. To solve f( x) > 0 using graphical methods, we find the set of values of x such that the graph of f( )yx= is above the x-axis. Similarly, to solve f(x) < 0, it is equivalent to finding the set of values of x such that the graph of y = f(x) is below the x-axis. For the next example, we will solve quadratic inequalities. ab0 1 a 1 b 0 ba<< 110 ab<<

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________________________ _______________________________ Chapter 3: Equations and Inequalities Page 4 of 20 Example 2 Without using a calculator, solve the following inequalities: (a) x 2 + 6x – 16 > 0, Solution: (a) x2 + 6x – 16 > 0 ( x + 8)(x – 2) > 0 x < –8 or x > 2 (b) –x 2 + 4x − 1 ≥ 0. Solution: (b) –x2 + 4x − 1 ≥ 0 Method 1: By using the quadratic formula Let –x2 + 4x − 1 = 0, i.e. x2 – 4x + 1 = 0 x = 4 16 4 2 ±− = 2 ± 3 Using graph to solve –x2 + 4x − 1 ≥ 0, 23 23 x− ≤≤+ Method 2: By completing the square 2 4 10xx− + −≥ ( ) ( ) ( ) 2 22 2 4 10 4 2 30 2 30 2 3 2 30 xx xx x xx − +≤ − + −≤ − −≤ −+ −− ≤ 23 23 x− ≤≤+ x y = –x2 + 4x − 1 2 – x 2 + –8 2 x 23−

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________________________ _______________________________ Chapter 3: Equations and Inequalities Page 5 of 20 (c) x2 + 2x + 1 ≤ 0. Solution: x2 + 2x + 1 ≤ 0 ( ) 2 10 1 x x +≤ =− Question: What is the solution if we are asked to solve x2 + 2x + 1 < 0 instead? No solution, so the solution set is or { }∅ , since there is no value of x that can satisfy the inequality. Question: How about solving x 2 + 2x + 1 ≥ 0 ? x can be any real number. The solution set is ( ) or , .−∞ ∞ – 1 x

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________________________ _______________________________ Chapter 3: Equations and Inequalities Page 6 of 20 1 x 0 −1 0 1 x −1 0 1 x 0 1 x 0 x Recap on how to sketch graphs of polynomials As seen from Example 2 above, solving inequalities involving po