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Explain why $\sqrt{x+3}<0$ has no solution in the set of real numbers while $\sqrt{x+3} \geq 0$ is true for all real numbers greater than or equal to $-3 .$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 8

Solving Radical Equations

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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first that consider this inequality. The square root of X plus three is less than zero. So this is gonna have no solution, No solution in the Rio numbers, right? So no solution for, um for X such that X is an element of the real numbers. This is the same. That right? There's no X. That's a real number. Such that X is a solution to this. Why? Well, what do we know about the square? Root was under the square root right? Cannot be negative. It's under the square root. Can be. If if this was negative. If X plus three was negative, then while we'd be undefined in the real numbers, right, we haven't imaginary number. So the domain of this, if it was a function, well, the smallest actually could possibly put in here would be negative. Three, right. Anything less than negative three. And you add three to it. You're gonna be negative. So the smallest execute possibly put in here. That makes sense. A two part of the domain would be negative. Three or negative. Three plus three is equal to zero. And anything larger than negative three, right? This is always gonna be great and zero So X plus three the square root of experts three can never be less than zero. Never look at this Inequality squared of experts. Three is greater than or equal to zero. This is true for all X, right, Because again, this small is actually possibly put in here. That's part of domain would be negative. Three, right? And native three is equal to zero. So this will be true and then any greater value, right? Any value of X greater than three while then the square root of X plus three is going to be positive. It's going to be great in zero. So then this would always be true for any possible acts you put in here any real number value of X. So the first inequality right is has no solution for any extra go number because the smallest actually could put in is negative. Three, right. Native three plus three is zero. So which is not less than zero, and any bigger X is not gonna be less than zero. But for any part, any possible acts in the domain again, the domain is while X must be, um, must be greater than or equal to negative three. Right? That's the domain. And then this second inequality right. The square root of X plus tree will always be greater than or equal to zero for all acts in the domain. All right, let's explain too much, but then they sense.

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