Solve the given rational equation for \( x \) over the interval \( [0, 2\pi] \): \( \frac{1}{x} = \sin(x) \)
Answer
From the graph, we find that the two curves intersect at approximately \( x = 1.114 \) and \( x = 2.03 \) within the given interval
Step 1 ：First, get rid of the fraction by multiplying both sides of the equation by \(x\): \( 1 = x \sin(x) \)
Step 2 ：Then, to isolate \(x\), set the equation equal to zero: \(x \sin(x) - 1 = 0\)
Step 3 ：This equation cannot be solved algebraically. So we use a graphing calculator or software to find the intersections of \(y = x \sin(x)\) and \(y = 1\) over the interval \( [0, 2\pi] \)
Step 4 ：From the graph, we find that the two curves intersect at approximately \( x = 1.114 \) and \( x = 2.03 \) within the given interval
