Use inspection to describe each inequality's solutionset. Do not solve any of the inequalities.??? I am so lost, any help would be greatly appreciated! (x-2)² ≤ 0

Use inspection to describe each inequality's solutionset. Do not solve any of the inequalities.??? I am so lost, any help would be greatly appreciated! (x-2)² ≤ 0

The answer is in the text below and in this YouTube short video: Solving Math in Seconds: Why (x-2)² ≤ 0 Has Only ONE Solution! #mathhacks

https://youtube.com/shorts/n-GohgjVuc4?feature=share

Understanding the Inequality's Solution Set Using Inspection

Let me help you understand how to describe the solution set for the inequality in your image without solving it.

The inequality shown is:

\[(x-2)^2 \leq 0\]

## Analyzing by Inspection

When we look at this inequality, we need to consider a key mathematical property: **a square of any real number is always greater than or equal to zero**.

That means:

- For any real number \[y\], we know that \[y^2 \geq 0\]

- The only time \[y^2 = 0\] is when \[y = 0\]

In our case, \[y = (x-2)\], so:

- \[(x-2)^2 = 0\] only when \[(x-2) = 0\], which happens when \[x = 2\]

- \[(x-2)^2 < 0\] is impossible for any real value of \[x\]

## Solution Set Description

The solution set for \[(x-2)^2 \leq 0\] is just the single point \[x = 2\].

This is because:

- The expression equals zero only when \[x = 2\]

- The expression is always positive (not ≤ 0) for any other value of \[x\]

Therefore, the solution set is \[\{2\}\] or simply \[x = 2\].