Linear Inequalities

1. Rules of Inequality

Inequalities behave like equations, with one major exception: Multiplying or Dividing by a negative number reverses the sign.

2. Solving Inequalities (One Variable)

We solve for \( x \) and represent the solution on a number line or in interval notation.

2.1 Linear Inequalities in Two Variables

A linear inequality in two variables is of the form:

Its solution is a region in the coordinate plane, not a single point.

Exam Tip: CBSE may ask to identify whether a given inequality is linear or not.

2.2 Inequalities Involving Fractions

While solving inequalities with fractions:

• Clear denominators carefully.
• If you multiply/divide by a negative denominator, reverse the inequality sign.

Common Mistake: Students forget to flip the sign.

2.3 Absolute Value Inequalities

Properties involving the modulus function (for \( a > 0 \)):

• \( |x| < a \implies -a < x < a \) (Distance is less than a)
• \( |x| \le a \implies -a \le x \le a \)
• \( |x| > a \implies x < -a \text{ or } x> a \) (Distance is greater than a)
• \( |x| \ge a \implies x \le -a \text{ or } x \ge a \)

3. Graphical Solution (Two Variables)

To solve \( ax + by > c \) graphically:

1. Replace inequality with \( = \) to get line equation.
2. Draw the line (Dotted for \( <,> \); Solid for \( \le, \ge \)).
3. Pick a test point (usually \( (0,0) \)) not on the line.
4. If True, shade towards the point. If False, shade away.

3.1 System of Linear Inequalities

When two or more linear inequalities are given together, the solution is the common region satisfying all inequalities.

The shaded region common to all inequalities is the solution.

Exam Tip: Shade carefully. Even one wrong boundary loses marks.

4. Solution Set & Verification

The solution set of an inequality is the set of all values that make the inequality true.

Note: Always substitute and check.

4.1 Word Problems (Applications)

Steps to formulate a linear inequality from a word problem:

1. Identify the unknown quantity and assign a variable (e.g., \( x \)).
2. Translate phrases like "at least" (\( \ge \)), "at most" (\( \le \)), "more than" (\( > \)), and "less than" (\( < \)) correctly.
3. Set up the inequality and solve.

One-Page Revision Checklist

Tick each box mentally. If you can explain it, you are exam-ready.

1. Basics of Inequalities

• Know symbols: <, >, =, =
• Inequality is different from equation
• Solution is a range of values, not one value

2. Rules of Inequalities

• Adding same number ? sign does NOT change
• Subtracting same number ? sign does NOT change
• Multiplying by positive number ? sign unchanged
• Multiplying or dividing by negative ? sign reverses

3. Solving Linear Inequalities (One Variable)

• Solve like equation till x is alone
• Reverse sign if needed
• Final answer written as inequality

4. Number Line Representation

• Open circle ? < or >
• Filled circle ? = or =
• Arrow direction correct
• Interval notation correct

5. Inequalities with Fractions

• Denominators cleared carefully
• Sign reversed if denominator is negative
• Final inequality checked

6. Linear Inequalities in Two Variables

• Standard form: ax + by </>/=/= c
• Represents a REGION, not a line
• Boundary line drawn correctly

7. Graphical Solution

• Replace inequality with equality
• Solid line ? = or =
• Dotted line ? < or >
• Test point (0,0) used correctly
• Correct side shaded

8. System of Linear Inequalities

• Each inequality shaded separately
• Final solution = COMMON shaded region
• Axes included when x = 0, y = 0

9. Solution Set & Verification

• Substitute value correctly
• LHS compared with RHS
• Final statement: satisfies / does not satisfy

10. Exam Smart Checks

• Sign reversal not forgotten
• Boundary line type correct
• Interval notation accurate
• Diagram neat and labeled