Final answer:
To find the equations of the tangent lines to the curve y = x - 1/x parallel to the line x - 2y = 4, you must find where the derivative of the curve is equal to the slope of the given line (0.5), and then use the point-slope form of the equation to find the tangent lines.
Step-by-step explanation:
The question involves finding the equations of the tangent lines to the curve y = x - 1/x that are parallel to a given line. To find these tangent lines, we first need to determine the slope of the given line by putting it into slope-intercept form (y = mx + b). The equation x - 2y = 4 can be rearranged to y = 0.5x - 2, so the slope (m) is 0.5. Next, we find the derivative of the curve to determine where on the curve the slope of the tangent lines is equal to 0.5. The derivative of y = x - 1/x is dy/dx = 1 + 1/x2. We set this equal to 0.5 and solve for x, which will give us the x-coordinate(s) of the point(s) of tangency. After finding the x-coordinate(s), we can use the original equation to find the corresponding y-coordinate(s). Finally, we use the point-slope form (y - y1 = m(x - x1)) with our found point(s) and the slope 0.5 to write the equation(s) of the tangent line(s).