3. Solve the following inequality and put your answer in interval notation. \[ \left|\frac{x+1}{2}-\frac{x-1}{3}\right|<1 \text {. } \] 4. Solve the equation \( \sqrt{x+2}+\sqrt{x-2}=\sqrt{4 x-2} \). S. .

### 3. Solving the Inequality We start with the inequality \[ \left|\frac{x+1}{2} - \frac{x-1}{3}\right| < 1. \] **Step 1.** Simplify the expression inside the absolute value: \[ \frac{x+1}{2} - \frac{x-1}{3} = \frac{3(x+1) - 2(x-1)}{6}. \] Expanding the numerator: \[ 3x + 3 - 2x + 2 = x + 5. \] Thus, \[ \frac{x+1}{2} - \frac{x-1}{3} = \frac{x+5}{6}. \] **Step 2.** Substitute back into the inequality: \[ \left|\frac{x+5}{6}\right| < 1. \] Multiply both sides by 6 (which is positive, so the inequality direction remains the same): \[ |x+5| < 6. \] **Step 3.** Rewrite the absolute value inequality as a compound inequality: \[ -6 < x+5 < 6. \] Subtract 5 from all parts: \[ -6 - 5 < x < 6 - 5 \quad \Longrightarrow \quad -11 < x < 1. \] **Answer in Interval Notation:** \[ (-11,\, 1) \] --- ### 4. Solving the Equation The equation to solve is \[ \sqrt{x+2}+\sqrt{x-2}=\sqrt{4 x-2}. \] **Step 1.** Determine the domain of the equation: - For \(\sqrt{x+2}\) we need \(x+2 \ge 0 \) \(\Longrightarrow x \ge -2\). - For \(\sqrt{x-2}\) we need \(x-2 \ge 0 \) \(\Longrightarrow x \ge 2\). - For \(\sqrt{4x-2}\) we need \(4x-2 \ge 0 \) \(\Longrightarrow x \ge \frac{1}{2}\). Thus, the domain is \(x \ge 2\). **Step 2.** Square both sides of the equation to eliminate the square roots: \[ \left(\sqrt{x+2}+\sqrt{x-2}\right)^2=\left(\sqrt{4x-2}\right)^2. \] Compute the left side: \[ (x+2)+(x-2)+2\sqrt{(x+2)(x-2)} = 2x+2\sqrt{x^2-4}. \] So the equation becomes: \[ 2x+2\sqrt{x^2-4} = 4x-2. \] **Step 3.** Isolate the square root term: \[ 2\sqrt{x^2-4} = 4x-2-2x = 2x-2. \] Divide by 2: \[ \sqrt{x^2-4} = x-1. \] **Step 4.** Square both sides again: \[ x^2-4 = (x-1)^2. \] Expand the right side: \[ x^2 - 4 = x^2 - 2x + 1. \] Subtract \(x^2\) from both sides: \[ -4 = -2x + 1. \] Solve for \(x\): \[ -2x = -4 - 1 = -5 \quad \Longrightarrow \quad x = \frac{5}{2}. \] **Step 5.** Verify the solution within the domain \(x \ge 2\). Substitute \(x=\frac{5}{2}\) into the original equation: \[ \sqrt{\frac{5}{2}+2}+\sqrt{\frac{5}{2}-2} = \sqrt{4\left(\frac{5}{2}\right)-2}. \] Simplify each term: \[ \sqrt{\frac{5}{2}+\frac{4}{2}}+\sqrt{\frac{5}{2}-\frac{4}{2}} = \sqrt{10-2}. \] \[ \sqrt{\frac{9}{2}}+\sqrt{\frac{1}{2}} = \sqrt{8}. \] \[ \frac{3}{\sqrt{2}}+\frac{1}{\sqrt{2}} = 2\sqrt{2}. \] \[ \frac{4}{\sqrt{2}} = 2\sqrt{2}. \] Multiply numerator and denominator by \(\sqrt{2}\): \[ \frac{4\sqrt{2}}{2} = 2\sqrt{2}. \] Thus, both sides are equal. **Final Answer:** \[ x=\frac{5}{2}. \]