HKDSE Mathematics Ronald Hui Tak Sun Secondary School
Book 5A Chapter 3 Compound Linear Inequalities in One Unknown
(i) x > 1 x 3 and (ii) x + 1 < 0 2x 1 When two or more linear inequalities are connected together by logical operators (e.g. ‘and’, ‘or’), we call it a compound linear inequality. Example: or
Solve the compound linear inequality ‘x > 1 and x 3’. Solving Compound Linear Inequalities Connected by ‘and’ x > 1 x 3 The hollow circle ‘ ’ means that ‘1’ is NOT included. They are the solutions of ‘x > 1’ and ‘x 3’ respectively. The solid circle ‘ ’ means that ‘3’ is included. But what are the solutions for ‘x > 1 and x 3’?
Values of x that can satisfy both the linear inequalities are the solutions. Solving Compound Linear Inequalities Connected by ‘and’ Solve the compound linear inequality ‘x > 1 and x 3’. x > 1 x 3 Let me show you the steps!
Graphical representation of x 3 Solve the compound linear inequality ‘x > 1 and x 3’. Step 1 Draw the graphical representations of the two linear inequalities on the same number line. Graphical representation of x > 1
All the values of x in the overlapping region satisfy both inequalities. Hence, it represents the solutions of the compound inequality. Step 2 Find the overlapping region of the two graphical representations. Solve the compound linear inequality ‘x > 1 and x 3’. Overlapping region
Step 2 Find the overlapping region of the two graphical representations. Solve the compound linear inequality ‘x > 1 and x 3’. Overlapping region ∴ The solutions can be represented graphically by: The solutions of ‘x > 1 and x 3’ are x 3.
(a) Solve ‘ and ’. Similarly, we can solve the following compound linear inequalities connected by ‘and’. x –2x < 3 Overlapping region ∴ The solutions of ‘x –2 and x < 3’ are –2 x < 3. Graphical representation:
(b) Solve ‘ and ’. x < –4x 4 No overlapping region for these two inequalities In this case, the compound inequality has no solutions.
In summary, we have: Compound inequalitySolutions x > a and x > b x > b x > a and x < b a < x < b x b no solutions Note:The solving steps are similar when the inequality signs ‘ ’ are replaced by ‘ ’ and ‘ ’ respectively.
Solve ‘x + 1 –2 and x 3 0’. When solving this compound inequality, we should solve each inequality separately first.
Solve ‘x + 1 –2 and x 3 0’.x + 1 –2 and x 3 0 x – (1) x (2) and ∵ x must satisfy both (1) and (2). ∴ The solutions of the compound inequality are x –3. Graphical representation:
Follow-up question Solve, and represent the solutions graphically. 5 – x < 4 2x + 1 7 2x 6 –x < –1 5 – x < 4 x > (1) x (2) and2x + 1 7 ∵ x must satisfy both (1) and (2). ∴ The solutions of the compound inequality are 1 < x 3. Graphical representation:
Summary on “AND” 22 October 2015 Ronald HUI