Introduction
HSA Notes brings short, clear, exam focused notes designed to help Class 9 SSC Part One Matric students score 90 plus marks in Punjab Board exams under the New Book 2025 to 2026 session. This page covers Unit 5, Linear Equations and Inequalities, from Exercise 5.1, Exercise 5.1 Review Exercise 5 including solving linear equations and linear inequalities.
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HSA Notes Features
- Short, easy to remember definitions.
- Step by step solutions for every question.
- Tricks and memory aids for faster revision.
- Exact alignment with the PCTB and PECTAA textbook.
- Equations and inequalities simplified into a clear, repeatable method.
- Past paper pattern questions included throughout.
- A fast revision format built for last minute review.
- Trusted by Punjab students.
Available Across All Punjab Boards and Top Schools
These notes are built for students across all Punjab Boards, including Lahore, Gujranwala, Rawalpindi, Faisalabad, Bahawalpur, Multan, Sargodha, Sahiwal, and DG Khan, and are used at top schools including The City School, Beaconhouse, Roots International, Lahore Grammar School, Karachi Grammar School, Froebel’s, The Educators, Army Public School, Divisional Public School, and Government Model School.
Exercise 5.1, Q1, Solving Equations
| No. | Equation | Solution |
|---|---|---|
| i | 12x plus 30 equals negative 6 | x equals negative 3 |
| ii | x over 3 plus 6 equals negative 12 | x equals negative 54 |
| iii | x over 2 minus 3x over 4 equals 1 over 12 | x equals negative one third |
| iv | 2 equals 7 times the quantity 2x plus 4, plus 12x | x equals negative 1 |
| v | the quantity 2x minus 1 over 3, minus 3x over 4, equals 5 over 6 | x equals negative 14 |
| vi | negative x over 2 equals 9 minus 2x | x equals 6 |
Exercise 5.1, Q2, Solving Inequalities
| No. | Inequality | Solution |
|---|---|---|
| i | x minus 6 is less than or equal to negative 2 | x is less than or equal to 4 |
| ii | negative 9 is greater than negative 16 plus x | x is less than 7 |
| iii | 3 plus 2x is greater than or equal to 3 | x is greater than or equal to 0 |
| iv | 6 times the quantity x plus 10, is less than or equal to 0 | x is less than or equal to negative 10 |
| v | 5x over 3, minus 3 over 4, is less than negative 1 over 12 | x is less than 2 over 5 |
| vi | one quarter x, minus one half, is less than or equal to negative 1 plus one half x | x is greater than or equal to 2 |
View the full PDF for complete working and number line diagrams for every inequality above.
How to Approach These Problems
For the equations in Q1, the general method is to clear any fractions first by multiplying through by the lowest common denominator, then collect all variable terms on one side and constant terms on the other, before dividing to isolate x.
This same approach applies to Q1(vi), where the equation negative x over 2 equals 9 minus 2x simplifies cleanly once both sides are multiplied by 2, giving negative x equals 18 minus 4x, which rearranges to 3x equals 18, and finally x equals 6.
For the inequalities in Q2, the method is nearly identical to solving equations, with one key difference, multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. This is why it is worth double checking your final inequality sign whenever a negative coefficient is involved, as seen in Q2(iv), where dividing by 6, a positive number, keeps the inequality direction unchanged.
Why This Unit Matters for Your Overall Score
Linear Equations and Inequalities brings together the algebraic manipulation skills built in earlier units and applies them to solving for an unknown value or a range of values.
Since these skills reappear constantly throughout the rest of the syllabus, from word problems to more advanced equation types in later units, building speed and accuracy here, particularly with fractional equations and inequalities, pays off across the entire Mathematics paper.
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Frequently Asked Questions
What is the main difference between solving an equation and solving an inequality? The method is very similar, but when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed, which is not a concern when solving equations.
How should I handle equations or inequalities with fractions? Multiply through by the lowest common denominator of all the fractions involved first, which clears the fractions and leaves a simpler equation or inequality to solve.
Are number line diagrams included for the inequalities? Yes, number line diagrams for every inequality in Q2 are included in the full downloadable PDF.
Are these notes aligned with the PCTB and PECTAA textbook? Yes, these notes are built to exactly match the PCTB and PECTAA New Book for Class 9 Mathematics for the 2025 to 2026 session.
Conclusion
With Exercise 5.1 fully solved, covering both linear equations and linear inequalities across twelve worked examples, Class 9 students have a strong foundation for Unit 5. Use these notes alongside regular practice with past papers to build the confidence needed for 90 plus marks in your Punjab Board exam.
Best of luck for your Class 9 Mathematics exam!