Inequalities — linear, solving, number line representation

Overview

Inequalities describe relationships where one quantity is greater than or less than another, rather than equal. They form the foundation for optimization, constraint problems, and understanding ranges of solutions.

Core Concepts

What is a Linear Inequality?

WHY linear? Because the variable $x$ appears to the first power only — no $x^2$, no $\sqrt{x}$, no $\frac{1}{x}$. The graph would be a straight line if we converted it to an equation.

The four symbols:

• $<$ : less than (strict, does NOT include the boundary)
• $\leq$ : less than or equal to (includes the boundary)
• $>$ : greater than (strict)
• $\geq$ : greater than or equal to (includes boundary)

Solving Linear Inequalities: The Rules

1. Add/Subtract freely: $x +3 < 5\implies x < 5 - 3 \implies x < 2$

   • WHY? Adding the same number to both sides preserves order (if $a < b$, then $a + k < b + k$)

2. Multiply/Divide by positive: $2x < 6 \implies x < 3$

   • WHY? Scaling both sides by a positive factor preserves order

3. Multiply/Divide by negative → FLIP the inequality sign: $-2x < 6 \implies x > -3$

   • WHY? When you multiply by a negative, you reverse the number line. If $2< 5$, then $-2 > -5$. This is the critical rule most students forget.

DERIVATION of the flip rule: Start with $a < b$. Multiply both sides by $-1$: $-a \quad ? \quad -b$ Take $a = 2, b = 5$: we have $2 < 5$, so $-2 \quad ? \quad -5$. On the number line, $-2$ is to the right of $-5$ (closer to zero), so $-2 > -5$. Therefore: multiplying by a negative reverses the inequality.

Solution: $3x - 7 \leq 11$ Add 7 to both sides: $3x \leq 18 \quad \text{(WHY? Preserves inequality)}$ Divide by 3 (positive): $x \leq 6 \quad \text{(WHY? Scaling by positive preserves order)}$

Answer: $x \in (-\infty, 6]$ (all real numbers up to and including 6)

Solution: $-4x + 5 > 17$ Subtract 5: $-4x > 12 \quad \text{(WHY? Standard subtraction rule)}$ Divide by $-4$ → FLIP the sign: $x < -3 \quad \text{(WHY? Dividing by negative reverses order)}$

Answer: $x \in (-\infty, -3)$ (all numbers less than $-3$)

Common trap: Students write $x > -3$ and forget to flip. Check: if $x = -2$, then $-4(-2) + 5 = 8 + 5 = 13 \not> 17$. Doesn't work! But $x = -4$ gives $-4(-4) + 5 = 16 + 5 = 21 > 17$. ✓

Solution: $\frac{2 - 3x}{5} \geq 4$ Multiply both sides by 5 (positive, no flip): $2 - 3x \geq 20 \quad \text{(WHY? Clearing denominator)}$ Subtract 2: $-3x \geq 18 \quad \text{(WHY? Isolating term with } x \text{)}$ Divide by $-3$ → FLIP: $x \leq -6 \quad \text{(WHY? Negative division reverses)}$

Answer: $x \in (-\infty, -6]$

Number Line Representation

WHY use a number line? Because it gives a visual map of all solutions at once. You instantly see the range and boundaries.

For $x \geq -1$:

• Closed circle at $-1$
• Arrow pointing right
• Interval notation: $[-1, \infty)$

Interval Notation Summary

Why it feels right: We're so used to equations where signs don't flip. Our brain treats division as "neutral" operation.

Steel-man: The student is correctly applying the equation rule but forgetting that inequalities have direction that reverses when you reflect the number line.

The fix: Always ask: "Am I multiplying/dividing by a negative?" If yes, flip the sign. The correct solution: $-2x < 8 \implies x > -4$.

Verification: Pick $x = 0$ (in our answer, since $0 > -4$): $-2(0) = 0$, and $0 < 8$ ✓. Pick $x = -5$ (outside our answer, since $-5 < -4$): $-2(-5) = 10$, and $10 < 8$ is false ✗ (correctly excluded). Confirms $x > -4$ is right.

Why it feels right: Students confuse $<$ with $\leq$.

The fix: $\leq$ and $\geq$ include the boundary → closed circle. $<$ and $>$ exclude it → open circle. Mnemonic: "equal" in the symbol = "filled in" circle.

If you buy $x$ chocolates, you spend $15x$ rupees. You need $15x \leq 100$ (spend less than or equal to your budget). Divide both sides by 15: $x \leq 6.67$. Since you can't buy part of a chocolate, $x \leq 6$ (you can buy 0, 1, 2, 3, 4, 5, or 6 chocolates).

Now imagine you have a debt of ₹50 and you're paying back ₹10 per week. After $x$ weeks, you've paid $10x$, so your debt is $50 - 10x$. When is your debt gone? When $50 - 10x \leq 0$. Solve: $50 \leq 10x \implies 5 \leq x$. You need at least 5 weeks.

The "flip rule" is like this: if you owe someone money (negative), and you multiply how much you owe, the more you owe, the worse your situation — the direction reverses!

Connections

• Linear Equations — inequalities are equations with "wigle room"
• Absolute Value Equations and Inequalities — combines inequality solving with distance concepts
• Systems of Inequalities — finding regions satisfying multiple constraints simultaneously
• Quadratic Inequalities — extends to curved boundaries, uses similar sign-analysis
• Linear Programming — optimization under linear inequality constraints
• Number Line and Real Numbers — the geometric structure underlying inequality representation

#flashcards/maths

What is a linear inequality in one variable? :: An inequality of the form $ax + b < 0$ (or $\leq, >, \geq$) where $a \neq 0$, whose solution is a set of values forming an interval

Concept Map

Hinglish (regional understanding)