True or False

One algebraic claim on the board — say, $\sqrt{x^2}=x$ — and one job: put it on trial. Is it ever true? Always true? An equation is not a fact just because it is written down; it is a claim about numbers, and claims earn their verdicts in court.

How we play

1. Vote first — true, false, or “it depends” — before any working.
2. Prosecute and defend: test values, rewrite, sketch, whatever bites.
3. Deliver a verdict with evidence: always, never, or exactly when.

The trial of $\sqrt{x^2}=x$

• “Try $x=3$: both sides are 3. Looks true.”
• “Try $x=-3$: the left side is $\sqrt{9}=3$, but the right side is $-3$. False.”
• “Graph both: $y=\sqrt{x^2}$ is a V, and $y=x$ is a line. They agree on the right half and split at the origin.”
• Verdict: true exactly when $x\ge 0$ — the square root forgot the sign, and only non-negative $x$ never had one to forget.

One variation

Claims that sound too strange to be true: "$2^{x+1}=2\cdot 2^x$" — sliding an exponential one step left is the same as doubling it, always, and the reason is an exponent law. That collision between shifting and stretching is one Transformations of Functions and The Exponential Function stage on purpose.

One witness is not a proof

A single counter-example kills an “always”. A single confirming example proves nothing — to speak about all numbers you need algebra or a graph. Testing a value you were not given is Checking Your Own Work wearing a courtroom robe.