Chapter 8, Problem 32CCSR

Solution

To explain: The equation $\sqrt{3x-4}+\sqrt{x+3}=-2$ has no real solution without solving the equation.

When we add two quantities under the square root sign the result obtained is either zero or positive.

Given:

The equation is $\sqrt{3x-4}+\sqrt{x+3}=-2$ .

Explanation:

Since the given equation is $\sqrt{3x-4}+\sqrt{x+3}=-2$ where right hand side of the equation states that by adding two quantities we obtain the result of $-2$ which is a negative number.

Now, in left hand side of the equation the two quantities added here are the square root of two quantities. And by property of square root, if $\left(n\right)$ is a real number then $\sqrt{n}\ge 0$ always by definition.

And from left hand side of equation, when we will add these two quantities, we will get either $0$ or a number greater than zero. But the result obtained in right hand side is negative from the given equation. Hence, the given equation $\sqrt{3x-4}+\sqrt{x+3}=-2$ has no real solution.

Conclusion:

When we add two quantities under the square root sign the result obtained is either zero or positive.