# How do you solve and graph #x^2+1<2x#?

We start with:

We'll solve this the same way as if it were an equal sign - so let's drop in an equal sign for now, so:

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To solve the inequality (x^2 + 1 < 2x):

- Subtract (2x) from both sides to get (x^2 - 2x + 1 < 0).
- Factor the quadratic expression to get ((x - 1)^2 < 0).
- Recognize that a square is always non-negative, so the expression ((x - 1)^2) is always greater than or equal to zero.
- Since we're looking for where ((x - 1)^2) is less than zero, there are no real solutions.

To graph the inequality:

- Plot the parabola (y = x^2 - 2x + 1).
- Since the expression ((x - 1)^2) is always greater than or equal to zero, the region where ((x - 1)^2 < 0) does not exist on the real number line.
- Therefore, the graph of the inequality (x^2 + 1 < 2x) is an empty set on the real number line.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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