Linear Inequalities — Lesson
1) Hook — A Fun Real-Life Example
Imagine you are organizing a cricket tournament in your school. You have a budget of ₹10,000 for buying cricket balls. Each ball costs ₹250. How many balls can you buy without exceeding your budget? This situation can be represented by a linear inequality, which helps you find the maximum number of balls you can purchase.
2) Core Concepts — Understanding Linear Inequalities
A linear inequality in one variable is similar to a linear equation but instead of an equal sign, it uses inequality symbols:
- < (less than)
- ≤ (less than or equal to)
- > (greater than)
- ≥ (greater than or equal to)
- ≠ (not equal to)
General form: ax + b < 0 or ax + b ≥ 0, where a and b are real numbers, and a ≠ 0.
Example 1:
Solve the inequality: 3x - 7 < 5
Solution:
| 3x - 7 < 5 |
| 3x < 5 + 7 |
| 3x < 12 |
| x < 4 |
Solution set: {x | x < 4}
Example 2: Graphical Representation
Graph the inequality x + 2 ≥ 5.
Solution:
- Rewrite as x ≥ 3.
- On the number line, mark the point 3.
- Since it is ≥ (greater than or equal), draw a solid dot at 3.
- Shade the region to the right of 3.
Interpretation: All values of x greater than or equal to 3 satisfy the inequality.
3) Key Formulas / Rules
- Rule 1: To solve ax + b < c, isolate x:
x < (c - b)/a if a > 0 - Rule 2: If a < 0, reverse the inequality sign when dividing by a:
ax + b < c ⇒ x > (c - b)/a - Rule 3: Inequalities can be added or subtracted on both sides without changing the inequality sign.
- Rule 4: Multiplying or dividing both sides by a positive number keeps the inequality sign same.
- Rule 5: Multiplying or dividing both sides by a negative number reverses the inequality sign.
4) Did You Know?
Linear inequalities are used in Linear Programming, a powerful tool in industries like manufacturing and agriculture in India to maximize profit or minimize cost. For example, farmers use inequalities to decide how much area to allocate to different crops under resource constraints like water and labor.
5) Exam Tips
- Always remember: When multiplying or dividing by a negative number, flip the inequality sign. This is a common error.
- Check your solution by substituting values from the solution set back into the inequality.
- Graphical representation questions often appear in board exams; practice drawing number lines with solid/dotted points.
- Previous year pattern: Questions like "Solve and represent on number line" and "Find solution set for inequalities" frequently appear.
- Practice word problems involving inequalities, especially related to budget, speed, and age — common in Indian contexts.
Linear Inequalities — Mcq
Linear Inequalities — Mnemonic
Mnemonic 1: "Inequality Direction Rule" 🔄
“Flip the sign when multiplying or dividing by negative!”
- Hindi rhyme: "Negative se guna karo, inequality palat jao!" 🔄
- Meaning: When you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign.
Mnemonic 2: "L.I.N.E." for solving Linear Inequalities 📏
- L - Look at the inequality carefully
- I - Isolate the variable on one side
- N - Note if multiplying/dividing by negative (flip sign!)
- E - Express the solution on number line or interval
Mnemonic 3: "INEQUAL" Funny Acronym 😄
- I - Ignore distractions, focus on variable
- N - Negative flips the sign (flip-flop!)
- E - Express answers as intervals or sets
- Q - Quickly check by substituting values
- U - Use number line for clarity
- A - Always keep inequality balanced
- L - Look for strict (< or >) or inclusive (≤ or ≥) signs
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