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Learn/direct proofs/Proving Inequalities

Lesson subsection

Proving Inequalities

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

Direct proofs with inequalities use basic facts like:

  • If a ≤ b and b ≤ c, then a ≤ c.
  • If a ≤ b and c ≥ 0, then ac ≤ bc.
  • Squares of real numbers are nonnegative: x² ≥ 0.

Example:

Prove: If x ≥ 2, then x² ≥ 4.

Assume x ≥ 2.
Since x ≥ 2 and x ≥ 2 ⇒ x·x ≥ 2·2 (multiplying by nonnegative numbers preserves order).
So x² ≥ 4.

Another example:

Prove: If 0 ≤ x ≤ 1, then x² ≤ x.

We can rewrite x² ≤ x as x² − x ≤ 0 ⇒ x(x − 1) ≤ 0.
When 0 ≤ x ≤ 1, we have x ≥ 0 and x − 1 ≤ 0, so their product is ≤ 0.

Key ideas:

  • Manipulate inequalities using allowed operations.
  • Sometimes rewrite the target inequality into a form like (something)² ≥ 0 or product of sign-known factors.

TL;DR — key idea

Inequality proofs use algebra plus basic order rules: preserve direction when multiplying by nonnegative numbers, and often rewrite to something obviously ≥ 0 or ≤ 0.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

  • 1

    Prove ONE of the following using a direct proof: 1. If x ≥ 3, then x² ≥ 9. 2. If 0 ≤ x ≤ 2, then x² ≤ 4x. Clearly indicate which inequality rules you are using (e.g., "multiplying both sides by a nonnegative number").

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.