To solve the given problem, let's denote [tex]\( y = x + \frac{1}{x} \)[/tex].
First, square both sides of this equation to involve [tex]\( x^2 \)[/tex]:
[tex]\[ y^2 = \left( x + \frac{1}{x} \right)^2 \][/tex]
Expanding the right-hand side, we get:
[tex]\[ y^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 \][/tex]
[tex]\[ y^2 = x^2 + 2 + \frac{1}{x^2} \][/tex]
According to the given equation:
[tex]\[ x^2 + \frac{1}{x^2} = 7 \][/tex]
Substitute this value into the expanded equation:
[tex]\[ y^2 = 7 + 2 \][/tex]
[tex]\[ y^2 = 9 \][/tex]
To find [tex]\( y \)[/tex], take the square root of both sides:
[tex]\[ y = \sqrt{9} \][/tex]
Since [tex]\( x > 0 \)[/tex], we have:
[tex]\[ y = 3 \][/tex]
Therefore, the value of [tex]\( x + \frac{1}{x} \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
