<p>Solve \(\left|\dfrac{x+2}{x-1}\right| = 2\).</p>
Step-by-Step Solution
Key Concept: Absolute value equations split into two cases: the expression equals 2 or equals -2. Each case yields a linear equation, but solutions must satisfy the domain restriction x ≠ 1.
<p><strong>Step 1:</strong> Write the two cases for |A| = 2</p><p>Case 1: $\frac{x+2}{x-1} = 2$</p><p>Case 2: $\frac{x+2}{x-1} = -2$</p><p><strong>Step 2:</strong> Solve Case 1</p><p>$\frac{x+2}{x-1} = 2$</p><p>$x + 2 = 2(x - 1)$</p><p>$x + 2 = 2x - 2$</p><p>$4 = x$</p><p><strong>Step 3:</strong> Solve Case 2</p><p>$\frac{x+2}{x-1} = -2$</p><p>$x + 2 = -2(x - 1)$</p><p>$x + 2 = -2x + 2$</p><p>$3x = 0$</p><p>$x = 0$</p><p><strong>Step 4:</strong> Verify domain restriction (x ≠ 1)</p><p>Both x = 4 and x = 0 satisfy x ≠ 1 ✓</p><p><strong>Step 5:</strong> Check solutions</p><p>For x = 4: $|\frac{4+2}{4-1}| = |\frac{6}{3}| = 2$ ✓</p><p>For x = 0: $|\frac{0+2}{0-1}| = |\frac{2}{-1}| = 2$ ✓</p><p>∴ <strong>Answer: x = 0 or x = 4</strong></p>
Correct Answer: x = 0, 4 or x = 1/3 (i.e., x = 0, 4, 7 depending on reading; answer: x = 0, 4, 7)