Inequalities are similar to equations, but instead of an equals sign (=), they use signs like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). Solving them is a key algebra skill.

1. The Golden Rule of Inequalities

Solving an inequality is almost identical to solving a regular equation, with one critical exception:

Whenever you multiply or divide both sides of an inequality by a NEGATIVE number, you MUST flip the inequality sign.

• > becomes <

• < becomes >

• ≥ becomes ≤

• ≤ becomes ≥

This is the most common place to lose marks, so always be careful!

Example:

• -2x > 10

• Divide by -2: x < -5 (The sign flips from > to <)

2. How to Solve Linear Inequalities (ax + b > cx + d)

Follow the same steps as you would for an equation:

1. Simplify: Expand any brackets.

2. Collect 'x' terms: Move all terms with 'x' to one side of the inequality (usually the left).

3. Collect constant terms: Move all the plain numbers to the other side.

4. Isolate 'x': Divide both sides by the coefficient of 'x' to get the final answer.

5. Check the Golden Rule: If you divided by a negative number in the final step, remember to flip the sign.

Example Walkthrough: Solve 3x - 8 ≤ 5x + 2

1. Move 'x' terms to the left: 3x - 5x - 8 ≤ 2 -2x - 8 ≤ 2

2. Move constant terms to the right: -2x ≤ 2 + 8 -2x ≤ 10

3. Isolate 'x' by dividing by -2 (This triggers the Golden Rule!): x ≥ -5 (The sign flips from ≤ to ≥)

3. Representing Solutions on a Number Line

This is a required skill and is very simple if you remember the difference between the circles:

• Open Circle (o): Used for < (less than) and > (greater than). It means the number itself is not included in the solution.

• Closed/Shaded Circle (•): Used for ≤ (less than or equal to) and ≥ (greater than or equal to). It means the number itself is included in the solution.

• Draw a number line.

• Mark the key number from your solution (e.g., -5).

• Draw the correct type of circle on that number.

• Draw a thick arrow in the direction of the solution.

Example:

• x > 3: Open circle on 3, arrow pointing to the right.

• x ≤ -1: Closed circle on -1, arrow pointing to the left.

Exam Tips & Common Pitfalls

• Pitfall 1: Forgetting to Flip the Sign. This is the #1 mistake. Always double-check your work if you divide or multiply by a negative.

• Pitfall 2: Using the Wrong Circle. Memorize the open vs. closed circle rule. It's an easy mark to get right or wrong.

• Exam Strategy for Word Problems:

  1. Read the problem carefully to identify the relationship (e.g., "mass of bag A is less than or equal to bag B").

  2. Define your variable (e.g., "Let x be the mass of one tea packet").

  3. Translate the word problem into a mathematical inequality.

  4. Solve the inequality using the steps above.

  5. Answer the specific question (e.g., "What is the maximum mass?"). If x ≤ 250, the maximum mass is 250g. If x ≥ 40, the minimum price is Rs. 40.