Inequalities

An inequality is a mathematical statement that compares two expressions using symbols like:

• $<$: Less than
• $>$: Greater than
• $\leq$: Less than or equal to
• $\geq$: Greater than or equal to
• $\neq$: Not equal to

Whereas an equation shows equality (e.g., $x + 2 = 5$), an inequality shows relative size or order (e.g., $x + 2 < 5$).

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Intuitive Understanding

Think of inequalities as number lines or balancing scales:

• If $x < 3$, you're looking at all values to the left of 3.
• If $x \geq -2$, you're looking at values on or to the right of -2.

It’s about ranges, not exact values.

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Key Concepts

Inequality Symbols and Their Meaning

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Solving Inequalities

Just like linear equations:

• Isolate the variable
• Simplify both sides

But with one extra rule:

When you multiply or divide both sides by a negative number, you must flip the inequality sign.

Example:

$$-2x > 6 \Rightarrow x < -3$$

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Rules to Remember

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Number Line Representation

Inequalities represent intervals on a number line.

• $x < 3$: Open circle at 3, shaded to the left
• $x \geq 2$: Closed circle at 2, shaded to the right

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Types of Solutions

1. Linear Inequality

$$2x - 3 < 5 \Rightarrow x < 4$$

2. Double Inequality

$$3 < 2x + 1 \leq 9 \Rightarrow \frac{2}{x} < 4 \Rightarrow x > 0.5$$

3. Compound Inequalities

Using AND / OR:

• $x < 3 \text{ and } x > 0$ ⟹ $0 < x < 3$
• $x < -2 \text{ or } x > 5$

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Inequalities in Word Problems

Example Types:

• "A number is more than 10 but less than 20"
  → $10 < x < 20$

• "At least ₹500"
  → $x \geq 500$

• "No more than 3 errors allowed"
  → $x \leq 3$

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Conceptual Tips

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Interval Notation

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Graphical Understanding

Imagine each inequality as a shaded region on a number line.

• Open circle (e.g., $<$, $>$) means endpoint excluded
• Closed circle (e.g., $\leq$, $\geq$) means endpoint included

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Examples

Example 1: Basic Linear Inequality

Q: Solve $5x - 7 < 18$
A:

$$5x < 25 \Rightarrow x < 5$$

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Example 2: Inequality with Negative Coefficient

Q: Solve $-3x + 4 \geq 10$
A:

$$-3x \geq 6 \Rightarrow x \leq -2$$

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Example 3: Compound Inequality

Q: Solve $1 < 2x - 3 < 5$
A:
Add 3 throughout:
$4 < 2x < 8 \Rightarrow 2 < x < 4$

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Example 4: Word Problem

Q: The speed limit is 60 km/h. A car must not exceed it. Express as inequality.
A:
$x \leq 60$

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Example 5: Inequality in Absolute Terms

Q: Solve $|x - 4| < 3$
A:
Split into two:

$$-3 < x - 4 < 3 \Rightarrow 1 < x < 7$$