The equation 2x = cos x has a solution between –2 and –1. Find this solution.

To solve the equation 2x = cos x between -2 and -1, we can use a numerical method such as the bisection method or Newton's method. Here, we'll use the bisection method.

Step 1: Define the function f(x) = 2x - cos x. We're looking for the x where f(x) = 0.

Step 2: Choose the interval [-2, -1]. We know that the solution lies in this interval.

Step 3: Calculate the midpoint m = (-2 - 1) / 2 = -1.5.

Step 4: Evaluate f(m). If f(m) is very close to 0, then m is the solution. If not, go to step 5.

Step 5: If f(-2) * f(m) < 0, then the solution lies in the interval [-2, m]. If f(m) * f(-1) < 0, then the solution lies in the interval [m, -1]. Choose the interval where the solution lies.

Step 6: Repeat steps 3 to 5 until you find the solution to the desired accuracy.

Note: This method requires that f(x) is continuous on the interval [-2, -1] and that f(-2) and f(-1) have opposite signs (i.e., f(-2) * f(-1) < 0). This ensures that there is at least one solution in the interval by the Intermediate Value Theorem.