Summary and Quick Revision

The Chapter in One Idea

Everything here flows from one asymmetry:

Inequalities obey equation rules — except that multiplying or dividing both sides by a negative number REVERSES the sign. Solve like an equation, watch for that one moment, and report the answer as an interval with the right brackets.

Hold that, and the rest of the chapter is bookkeeping: which circle to draw, which bracket to write, which parts of a chain to operate on.

Rules and Notation — Formula Card

• Rule 1: add or subtract any number on both sides — sign unchanged.
• Rule 2: multiply or divide by a positive number — sign unchanged; by a negative number — sign reverses ($<$ to $>$, $\le$ to $\ge$).
• Strict vs slack: $<, >$ exclude the boundary; $\le, \ge$ include it.
• Interval notation: $(a, b)$ open, $[a, b]$ closed, $[a, b)$ and $(a, b]$ half-open; $x > a$ is $(a, \infty)$, $x \le b$ is $(-\infty, b]$. Round bracket = excluded, square = included; $\infty$ always gets a round bracket.
• Translation: "at most" $\to \le$; "at least" $\to \ge$; "exceeds/more than" $\to >$; "below/less than" $\to <$; "between $a$ and $b$" $\to a < x < b$.
• Solution and solution set: any value making the statement true; the set of all of them. The answer depends on the allowed domain — naturals, integers or reals.

Number Lines, Chains and Systems — Formula Card

• Graphing: strict sign → open circle at the boundary; slack sign → filled circle; darken the side where solutions lie.
• Double inequality: operate on all three parts together: $-8 \le 5x - 3 < 7$ becomes $-5 \le 5x < 10$, then $-1 \le x < 2 = [-1, 2)$.
• Negative divisor in a chain: BOTH signs flip and the chain reverses — then rewrite smallest-to-largest: $-15 \le -3x \le 11$ gives $-\frac{11}{3} \le x \le 5$.
• System of inequalities: solve each separately, then intersect — keep only the overlap of the dark lines. Empty overlap = no solution ($\varnothing$), a legitimate answer.
• "And" vs "or": simultaneous conditions intersect ($\cap$); alternative conditions unite ($\cup$).
• Word-problem recipe: name the variable → translate → solve → interpret (trim to marks $\le 100$, positive lengths, whole counts; consecutive odds/evens are $x, x+2$; mixtures track litres of pure substance: $p\%$ of $V$ litres is $\frac{p}{100}V$).

JEE Quick Card — Modulus, Rational, Wavy Curve

Modulus inequalities ($a > 0$):

• $|x| < a \Leftrightarrow -a < x < a$ (band); $|x| \le a \Leftrightarrow -a \le x \le a$.
• $|x| > a \Leftrightarrow x < -a$ or $x > a$ (two rays, a union).
• $|x - c| < r \Leftrightarrow c - r < x < c + r$ — distance from $c$ below $r$.
• Degenerate: $|f(x)| <$ (negative) is $\varnothing$; $|f(x)| >$ (negative) is $\mathbb{R}$.

Rational inequalities: never multiply across by an expression of unknown sign. Bring everything to one side as $\frac{P(x)}{Q(x)}$, mark critical points, sign-analyse.

Wavy curve: rightmost region positive (for positive leading coefficients); sign alternates at each odd-power factor, stays at even-power factors. Numerator zeros join the answer only under $\le$/$\ge$; denominator zeros never do.

Quadratic inequalities: $(x - \alpha)(x - \beta) > 0$ (with $\alpha < \beta$) solves outside the roots, $< 0$ between them.

Domains as systems: $\sqrt{f(x)}$ needs $f(x) \ge 0$; several radicals intersect their conditions.

Last-Minute Mistake Checklist

Before the exam, scan this list — each item is a real mark lost by real students:

1. Dividing by a negative and not flipping the sign — the chapter's number-one error ($-2x < 4$ gives $x > -2$, not $x < -2$).
2. Flipping when you shouldn't: adding or subtracting never flips, and neither does dividing by a positive.
3. Wrong bracket: strict sign with a square bracket or slack with a round one — $x \ge 8$ is $[8, \infty)$, and $\infty$ never takes a square bracket.
4. Open vs filled circle mismatched to the sign on number-line graphs.
5. In a chain, operating on only two of the three parts — every move hits all three.
6. After dividing a chain by a negative: flip both signs AND rewrite in increasing order ($5 \ge x \ge -\frac{11}{3}$ becomes $-\frac{11}{3} \le x \le 5$).
7. Solving a system with union instead of intersection — "and" means overlap.
8. Forgetting the domain: naturals exclude 0 and negatives; "consecutive odd" steps by 2, not 1.
9. Cross-multiplying a rational inequality by $(x - a)$ — sign unknown; move everything to one side instead. And never include a denominator zero.
10. Word problems: answering with the raw interval and skipping the interpretation — marks cap at 100, lengths are positive, counts are whole numbers.