Solution: Solve $2x = 60^\circ, 120^\circ, 420^\circ, 480^\circ$ within $0^\circ \leq 2x \leq 720^\circ$. Thus, $x = 30^\circ, 60^\circ, 210^\circ, 240^\circ$. Sum: $30 + 60 + 210 + 240 = 540^\circ$.
["How to Solve the Equation $2x = 60^\circ, 120^\circ, 420^\circ, 480^\circ$ — Step-by-Step Guide", "Solving trigonometric or linear equations within a specific range is a common task in mathematics. This article focuses on solving the equation $2x = 60^\circ, 120^\circ, 420^\circ, 480^\circ$ with the constraint $0^\circ \leq 2x \leq 720^\circ$, and then summing the valid solutions for $x$.", "---", "### Understanding the Problem", "We are given the equation:", "$$\n2x = 60^\circ, \ 120^\circ, \ 420^\circ, \ 480^\circ\n$$", "and the range for $2x$ is:", "$$\n0^\circ \leq 2x \leq 720^\circ\n$$", "Our goal is to find all values of $x$ that satisfy these conditions.", "---", "### Step 1: Divide both sides by 2", "To isolate $x$, divide every term by 2:", "$$\nx = \frac{60^\circ}{2},\ \frac{120^\circ}{2},\ \frac{420^\circ}{2},\ \frac{480^\circ}{2}\n$$", "This simplifies to:", "$$\nx = 30^\circ,\ 60^\circ,\ 210^\circ,\ 240^\circ\n$$", "---", "### Step 2: Verify the range condition $0^\circ \leq 2x \leq 720^\circ$", "Check each $2x$ value:", "- $60^\circ$ → valid\n- $120^\circ$ → valid\n- $420^\circ$ → valid (within range)\n- $480^\circ$ → valid (within range)", "All values satisfy the domain condition.", "---", "### Step 3: Sum the valid $x$ values", "Now compute the sum of the solutions:", "$$\n30^\circ + 60^\circ + 210^\circ + 240^\circ = 540^\circ\n$$", "---", "### Why This Approach Works", "By solving for $2x$ first, we simplify the linear equation. Then, dividing ensures all $x$ values remain within realistic bounds under the given constraint. The range check is crucial when dealing with periodic functions like trigonometric equations, but here it simply confirms valid inputs.", "---", "### Final Summary", "- Values of $2x$: $60^\circ, 120^\circ, 420^\circ, 480^\circ$\n- Corresponding $x$: $30^\circ, 60^\circ, 210^\circ, 240^\circ$\n- Total sum: $30 + 60 + 210 + 240 = 540^\circ$", "This method efficiently resolves equations with restricted domains, ensuring both accuracy and clarity.", "---", "Keywords: solve $2x = 60^\circ, 120^\circ, 420^\circ, 480^\circ$, $x$ values, sum of solutions, step-by-step equation solving, trigonometric equations, linear equations range, $0^\circ \leq 2x \leq 720^\circ$", "---", "Whether you're a student studying math, or a curious learner exploring equations, mastering this step-by-step process helps build strong problem-solving skills."]
