Lesson 8 of 12 · Algebra Foundations
Inequalities & AM-GM
Linear and quadratic inequalities are mechanical once you're careful about sign flips. AM-GM is different — it's the single fastest tool for "minimize a sum" or "maximize a product" problems, and its equality case often pins down the answer directly, without any calculus.
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Learning objectives
- Solve linear inequalities correctly, flipping the direction whenever multiplying or dividing by a negative.
- Solve quadratic inequalities using the roots as boundaries and sign analysis between them.
- Apply the AM-GM inequality to find the minimum of a sum or the maximum of a product under a constraint.
- Recognize AM-GM's equality condition and use it to identify exactly where the optimum occurs.
The core idea
Linear inequalities behave exactly like linear equations, with one exception: multiplying or dividing both sides by a negative number flips the inequality's direction. Quadratic inequalities reduce to the same factoring skill from earlier lessons — find the roots, then test which intervals between them satisfy the original inequality.
AM-GM is a different kind of tool entirely. It converts "find the minimum of a sum" or "find the maximum of a product" into a direct algebraic bound, and tells you exactly when that bound is achieved — no derivatives required.
AM-GM inequality (two terms)
(a + b)/2 ≥ √(ab), for a, b ≥ 0
Equality holds exactly when a = b. This single fact is enough to both bound a sum from below and identify where that bound is achieved.
AM-GM inequality (general)
(a₁+a₂+⋯+aₙ)/n ≥ ⁿ√(a₁a₂⋯aₙ)
Same idea with more terms — equality when all the terms are equal.
Quadratic inequality via sign analysis
For (x−r)(x−s) with r < s: the product is positive outside [r, s] and negative between r and s. Find the roots first, then test one point in each region (or reason directly from the sign pattern).
Worked example 1 — Linear inequality with a sign flip
Problem
Solve 3 − 2x ≥ 7.
Key insight
Isolating x requires dividing by −2, which flips the inequality direction.
Solution
−2x ≥ 4 → dividing by −2 and flipping: x ≤ −2.
Takeaway
The flip is easy to forget under time pressure — a quick sanity check (plug in x=0) catches it immediately if you forgot.
Worked example 2 — Quadratic inequality
Problem
Solve x² − 5x + 6 > 0.
Key insight
Factor first to find the boundary points, then determine the sign in each region.
Solution
(x−2)(x−3) > 0. The roots 2 and 3 split the number line into three regions; testing a point in each shows the product is positive outside [2, 3]: x < 2 or x > 3.
Takeaway
You only need to test one point per region — the sign alternates as you cross each root.
Worked example 3 — AM-GM to minimize a sum
Problem
For x > 0, find the minimum value of x + 4/x.
Key insight
Apply AM-GM directly to the two terms x and 4/x.
Solution
(x + 4/x)/2 ≥ √(x · 4/x) = √4 = 2, so x + 4/x ≥ 4. Equality holds when x = 4/x, i.e. x = 2. Minimum value: 4.
Takeaway
The product of the two terms (x)(4/x) = 4 is constant — that's exactly the setup AM-GM is built for.
Worked example 4 — AM-GM to maximize a product
Problem
If x + y = 10 with x, y > 0, find the maximum value of xy.
Key insight
This time the sum is fixed, so AM-GM bounds the product from above instead.
Solution
(x+y)/2 ≥ √(xy) → 5 ≥ √(xy) → xy ≤ 25, with equality at x = y = 5. Maximum value: 25.
Takeaway
Fixed sum → bounds the product (maximum). Fixed product → bounds the sum (minimum). Recognizing which one you're given tells you which direction the bound goes.
Strategy notes
Fixed sum bounds the product; fixed product bounds the sum
This is the entire skill of recognizing an AM-GM setup. If a problem fixes a sum and asks for a maximum product, or fixes a product and asks for a minimum sum, AM-GM is almost certainly the intended tool.
Always state the equality condition
A bound alone ("the sum is at least 4") isn't a complete answer to a minimize/maximize problem — you also need to confirm that equality is actually achievable within the problem's constraints.
Common mistakes
Forgetting to flip the inequality
Multiplying or dividing both sides of an inequality by a negative number reverses its direction. This is the single most common error in this entire topic.
Getting the quadratic inequality direction backwards
For (x−r)(x−s) with r < s, the product is negative between the roots and positive outside them — mixing these up is easy without sketching (even mentally) the parabola's shape.
Applying AM-GM without checking the terms are nonnegative
AM-GM requires every term to be nonnegative. Applying it to expressions that could be negative (without first verifying the sign) produces a bound that may not actually hold.
Practice
20 questions across four difficulty tiers. Linear-inequality questions ask for a single boundary value. Quadratic-inequality questions ask for both boundary values, comma-separated. AM-GM questions ask for the optimal (minimum or maximum) value itself. Up to three tries per question before the solution is shown.