Chapter 22 — Inequalities
Table of Contents
22.1 Manipulating Inequalities
When we move from equations to expressions where two sides are not necessarily equal, we enter the realm of inequalities.
Strict vs. Non-strict Inequalities
- Strict: $>$ or $<$ — the two sides are definitely not equal.
- Non-strict: $\geq$ or $\leq$ — the two sides may also be equal.
For example, $1 \geq 0$ is true even though $1 \neq 0$.
Rules for Inequalities
| Operation | Rule |
|---|---|
| Add/Subtract same quantity | Sign preserved: $x < y \Rightarrow x + c < y + c$ |
| Multiply/Divide by positive | Sign preserved: $x < y,\; c>0 \Rightarrow cx < cy$ |
| Multiply/Divide by negative | Sign reverses: $x < y,\; c<0 \Rightarrow cx > cy$ |
| Multiply by $0$ | Illegal (yields $0$ on both sides) |
| Adding inequalities (same direction) | $a < c, \; b < d \Rightarrow a+b < c+d$ |
| Powers (positive quantities) | $x > y > 0 \Rightarrow x^n > y^n$ for $n > 0$ |
Example 22-1. Taking reciprocals with inequalities:
- If both sides are positive: $2 < 3 \Rightarrow \frac{1}{2} > \frac{1}{3}$ — sign reverses.
- If one side is negative, one positive: $-2 < 3 \Rightarrow -\frac{1}{2} < \frac{1}{3}$ — sign preserved.
- If both sides are negative: $-3 < -2 \Rightarrow -\frac{1}{3} > -\frac{1}{2}$ — sign reverses.
Rule: The inequality sign reverses when reciprocals are taken if both sides have the same sign.
Exercise 22-1. Convince yourself that adding/subtracting the same thing and adding same-direction inequalities is valid. Try examples.
Exercise 22-2. For what restrictions on $x$ and $y$ is it true that $x > y \Rightarrow -x < y$?
Exercise 22-3. What happens when reciprocals are taken and both sides are negative?
22.2 Linear Inequalities
Linear inequalities contain only first powers of the variables, e.g. $x > 4y + 3$.
Two-variable linear inequalities
Consider $x > 4y + 3$, equivalently $x - 4y - 3 > 0$.
- Graph the boundary line $x - 4y - 3 = 0$ (draw it dashed for strict, solid for non-strict).
- The solution region is an entire half-plane on one side of the line.
- To determine which side: test a point (e.g. $(0,0)$). If $0 > 4(0) + 3$ is false, shade the other side.
Exercise 22-4. Draw the lines $x - 4y - 3 = k$ for $k = -4, -2, 0, 2, 4$ on one graph.
Exercise 22-5. Draw the solution of $x + y > 0$.
Exercise 22-6. Draw the solution of $x - y < 0$ on the same graph.
Exercise 22-7. Draw the solution of the simultaneous system $x + y > 0$, $x - y < 0$.
Exercise 22-8. Show how the solution region changes if either or both strict inequalities become non-strict.
One-variable linear inequalities
Example 22-2. $x < -3$ is graphed on a number line with an open circle at $-3$ (since $-3$ is excluded). With $\leq$ we use a filled circle.
22.3 Quadratic Inequalities
Quadratic inequalities in one variable (e.g. $x^2 - x - 6 > 0$) require sign analysis.
Strategy — Sign Analysis:
- Factor the expression: $x^2 - x - 6 = (x-3)(x+2)$.
- Find the zeros: $x = 3$ and $x = -2$.
- Determine the sign of each factor in each interval.
- Combine: a product is positive when factors have the same sign, negative when they differ.
Worked Example. Solve $(x-3)(x+2) > 0$.
| Interval | $x-3$ | $x+2$ | Product |
|---|---|---|---|
| $x < -2$ | $-$ | $-$ | $+$ ✓ |
| $-2 < x < 3$ | $-$ | $+$ | $-$ ✗ |
| $x > 3$ | $+$ | $+$ | $+$ ✓ |
Solution: $x < -2$ or $x > 3$, i.e. $(-\infty, -2) \cup (3, \infty)$.
For $\geq$, include the endpoints: $(-\infty, -2] \cup [3, \infty)$.
Interval Notation
| Notation | Meaning |
|---|---|
| $[a, b]$ | $a \leq x \leq b$ (endpoints included) |
| $(a, b)$ | $a < x < b$ (endpoints excluded) |
| $[a, b)$ | $a \leq x < b$ (left included, right excluded) |
| $(-\infty, a) \cup (b, \infty)$ | $x < a$ or $x > b$ |
Exercise 22-9. Plot the solution to $x^2 + 5x + 6 < 0$.
Exercise 22-10. Plot the solutions to $x^2 - 6x + 9 > 0$ and $x^2 - 6x + 9 < 0$.
Exercise 22-12. Solve $x^4 - 5x^2 + 6 > 0$ by letting $y = x^2$ and considering the signs of the factors.
Warning — Inequalities with fractions. For $\dfrac{2}{x+1} > \dfrac{3}{x-3}$, do not cross-multiply (the sign of $x-3$ is unknown). Instead:
- Move everything to one side: $\dfrac{2}{x+1} - \dfrac{3}{x-3} \geq 0$.
- Combine: $\dfrac{-x - 9}{(x+1)(x-3)} \geq 0$.
- Factor and do sign analysis on the three linear factors.
Solution: $(-\infty, -9] \cup (-1, 3)$. Note: $x = -9$ is included (makes numerator $0$), but $x = -1, 3$ are excluded (make denominator $0$).
22.4 Absolute Value Inequalities
Key equivalences:
\[|M| < a \iff -a < M < a \qquad (a > 0)\]\(|M| > a \iff M > a \;\text{ or }\; M < -a\)
| Worked Example. Solve $4 | x + 3 | + 3 > 9$. |
-
Isolate: $ x + 3 > \dfrac{3}{2}$. - Split: $x + 3 > \dfrac{3}{2}$ or $x + 3 < -\dfrac{3}{2}$.
- Solve: $x > -\dfrac{3}{2}$ or $x < -\dfrac{9}{2}$.
In interval notation: $\left(-\infty, -\dfrac{9}{2}\right) \cup \left(-\dfrac{3}{2}, \infty\right)$.
When the absolute value is on the small side:
\[|x + 3| < \frac{3}{2} \implies -\frac{3}{2} < x + 3 < \frac{3}{2} \implies -\frac{9}{2} < x < -\frac{3}{2}.\]Exercise 22-13. Plot the solution to $4|x+3| + 3 > 9$ and express it in interval notation.
22.5 The Trivial Inequality
The Trivial Inequality. For any real number $x$:
\[x^2 \geq 0\]Equality holds if and only if $x = 0$.
Despite appearing content-free, this is surprisingly powerful.
Example 22-5. Prove that for all real $x, y$: $\dfrac{x^2 + y^2}{2} \geq xy$.
Proof. Multiply both sides by $2$: need $x^2 + y^2 \geq 2xy$. Rearrange: $x^2 - 2xy + y^2 \geq 0$, i.e. $(x - y)^2 \geq 0$. ✓ (by the Trivial Inequality)
Each step is reversible, so the original inequality holds.
Application — Minimizing quadratics. To find the minimum of $x^2 + 5x + 1$, complete the square:
\[x^2 + 5x + 1 = \left(x + \frac{5}{2}\right)^2 - \frac{21}{4}.\]Since the squared term $\geq 0$, the minimum value is $-\dfrac{21}{4}$, attained at $x = -\dfrac{5}{2}$.
Exercise 22-14. In the proof of Example 22-5, show that each step is reversible (the true statement at the end implies the result at the start).
22.6 AM-GM Inequality
Arithmetic Mean – Geometric Mean (AM-GM) Inequality.
For non-negative real numbers $a$ and $b$:
\[\frac{a + b}{2} \geq \sqrt{ab}\]Equality holds if and only if $a = b$.
General form for $n$ non-negative reals $a_1, a_2, \ldots, a_n$:
\(\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}\)
Proof of the 2-variable case using the Trivial Inequality:
\((\sqrt{a} - \sqrt{b})^2 \geq 0 \implies a - 2\sqrt{ab} + b \geq 0 \implies \frac{a+b}{2} \geq \sqrt{ab}. \quad \square\)
AM-GM is one of the most important tools in competition mathematics for optimizing expressions and establishing bounds. While the full treatment appears in Volume 2, the core idea is:
The arithmetic mean of non-negative numbers always exceeds (or equals) their geometric mean.
22.7 Cauchy-Schwarz
Cauchy-Schwarz Inequality. For real numbers $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$:
\[(a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2) \geq (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n)^2\]Equality holds if and only if $\dfrac{a_i}{b_i}$ is constant for all $i$ (or some $b_i = 0$ with the corresponding $a_i = 0$).
Two-variable version:
\[(a_1^2 + a_2^2)(b_1^2 + b_2^2) \geq (a_1 b_1 + a_2 b_2)^2.\]This can be proven by expanding both sides and applying the Trivial Inequality to $(a_1 b_2 - a_2 b_1)^2 \geq 0$.
The full treatment of Cauchy-Schwarz and its many applications appears in Volume 2 of AoPS.
Problems to Solve for Chapter 22
Problem 414. Describe all $y$ such that $3y > 4 - y$ and $-2y > 1 + y$.
Problem 415. Find all $z$ such that $\dfrac{2}{z} > 1$.
Problem 416. Find all $x$ such that $x^2 + x - 30 > 0$ and $\dfrac{6}{x} > 0$.
Problem 417. A ball thrown upward from a tower has height $-t^2 + 60t + 700$. What is the greatest height the ball attains?
Problem 418. Find all $x$ satisfying $\sqrt{x} < 2x$. (AHSME 1980)
Problem 419. If $1.2 < a < 5.1$ and $3 < b < 6$, find the highest possible value of $a/b$ as a decimal. (MATHCOUNTS 1988)
Problem 420. How many integers satisfy $|x| + 1 > 3$ and $|x - 1| < 3$? (MATHCOUNTS 1987)
Problem 421. For how many integers $n$ is $3 - \dfrac{2}{n} < 3$? (MATHCOUNTS 1992)
Problem 422. If $x < a < 0$, prove that $x^2 > ax > a^2$. (AHSME 1956)
Problem 423. Show that if $x - y > x$ and $x + y < y$, then both $x$ and $y$ must be negative. (AHSME 1966)
Problem 424. Given positive integers $a, b, c, d$ satisfying $\dfrac{b}{a} < \dfrac{d}{c} < 1$, arrange in increasing order:
\[\frac{b}{a}, \quad \frac{d}{c}, \quad \frac{bd}{ac}, \quad \frac{b+d}{a+c}\](MAΘ 1987)
Problem 425. Solve for $x$: $x + \sqrt{x} < -2$. (MAΘ 1990)
Problem 426. If the smallest value of $y = 3x^2 + 6x + k$ is $4$, find $k$. (MATHCOUNTS 1989)
Problem 427. If $r > 0$, then for all $p, q$ with $pq \neq 0$ and $pr > qr$, which must be true?
A. $-p > -q$ B. $-p > q$ C. $1 > -q/p$ D. $1 < q/p$ E. none of these
Problem 428. Find the greatest integer $x$ for which $3^{20} > 3^{2x}$. (MAΘ 1991)