Question

Which inequality represents all values of x for which the product below is defined?√5x * √x+3A. x>0B. x≤-3C. x≥-3D. x≥0

Answer 1

Having the following product:

`(\sqrt[]{5x})\cdot(\sqrt[]{x+3})`

We need to establish the values of x for which it is defined.

The product has 2 square roots. We know that square roots are defined in real numbers only when its argument is 0 or a positive number.

We can begin checking the first one: √(5x). The argument of the square root (5x) has to be equal or larger than 0, then:

`\begin{gathered} 5x\ge0 \\ \end{gathered}`

We can divide both sides by 5, having:

`x\ge0`

The first factor is defined for values of x larger or equal to 0.

Following the same logic for the other factor:

`\begin{gathered} x+3\ge0 \\ x\ge-3 \end{gathered}`

Then, we have two restrictions so far:

`\begin{gathered} x\ge0 \\ x\ge-3 \end{gathered}`

The restrictions for the product of both radicals will be the intersection of the two previous conditions. The first restriction goes from 0 to infinity and the second one from -3 to infinity. The intersection of both intervals gives the numbers from 0 to infinity. Then, the values of x for which the product is defined are:

`x\ge0`

Correct option is D.