Step-by-Step Solution
Key Concept: Recognize that |2^x - 1| + |2^x + 1| represents the sum of distances from 2^x to two points (1 and -1) on the number line. Since 2^x > 0 always, we have 2^x + 1 > 0, so |2^x + 1| = 2^x + 1. The equation simplifies by analyzing the sign of (2^x - 1).
<p><strong>Step 1:</strong> Observe that 2^x > 0 for all real x. Therefore, 2^x + 1 > 0 always, so |2^x + 1| = 2^x + 1.</p><p><strong>Step 2:</strong> The equation becomes |2^x - 1| + (2^x + 1) = 2, which simplifies to |2^x - 1| = 1 - 2^x.</p><p><strong>Step 3:</strong> For |2^x - 1| = 1 - 2^x to be valid, we need 1 - 2^x ≥ 0 (absolute value is non-negative), giving 2^x ≤ 1.</p><p><strong>Step 4:</strong> When 2^x ≤ 1, we have 2^x - 1 ≤ 0, so |2^x - 1| = -(2^x - 1) = 1 - 2^x, which is exactly our equation. This confirms consistency.</p><p><strong>Step 5:</strong> Solve 2^x ≤ 1: Since 2^0 = 1 and 2^x is increasing, 2^x ≤ 1 gives x ≤ 0.</p><p>∴ <strong>Answer: x ≤ 0</strong></p>
Correct Answer: x ≤ 0