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. .

Solve the inequality by following steps: - step0: Solve for \(x\): \(\frac{x+1}{2}-\frac{x-1}{3}<1\) - step1: Multiply both sides: \(\left(\frac{x+1}{2}-\frac{x-1}{3}\right)\times 2\times 3<1\times 2\times 3\) - step2: Multiply the terms: \(3x+3-2x+2<6\) - step3: Simplify: \(x+5<6\) - step4: Move the constant to the right side: \(x<6-5\) - step5: Subtract the numbers: \(x<1\) Solve the equation \( \frac{x+1}{2}-\frac{x-1}{3}>-1 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\frac{x+1}{2}-\frac{x-1}{3}>-1\) - step1: Multiply both sides: \(\left(\frac{x+1}{2}-\frac{x-1}{3}\right)\times 2\times 3>-2\times 3\) - step2: Multiply the terms: \(3x+3-2x+2>-6\) - step3: Simplify: \(x+5>-6\) - step4: Move the constant to the right side: \(x>-6-5\) - step5: Subtract the numbers: \(x>-11\) Solve the equation \( \frac{x+1}{2}-\frac{x-1}{3}>-1 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\frac{x+1}{2}-\frac{x-1}{3}>-1\) - step1: Multiply both sides: \(\left(\frac{x+1}{2}-\frac{x-1}{3}\right)\times 2\times 3>-2\times 3\) - step2: Multiply the terms: \(3x+3-2x+2>-6\) - step3: Simplify: \(x+5>-6\) - step4: Move the constant to the right side: \(x>-6-5\) - step5: Subtract the numbers: \(x>-11\) Solve the equation \( \sqrt(x+2)+\sqrt(x-2)=\sqrt(4x-2) \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{x+2}+\sqrt{x-2}=\sqrt{4x-2}\) - step1: Find the domain: \(\sqrt{x+2}+\sqrt{x-2}=\sqrt{4x-2},x\geq 2\) - step2: Swap the sides: \(\sqrt{4x-2}=\sqrt{x+2}+\sqrt{x-2}\) - step3: Raise both sides to the \(2\)-th power\(:\) \(\left(\sqrt{4x-2}\right)^{2}=\left(\sqrt{x+2}+\sqrt{x-2}\right)^{2}\) - step4: Evaluate the power: \(4x-2=2x+2\sqrt{x^{2}-4}\) - step5: Move the expression to the left side: \(4x-2-2x=2\sqrt{x^{2}-4}\) - step6: Calculate: \(2x-2=2\sqrt{x^{2}-4}\) - step7: Swap the sides: \(2\sqrt{x^{2}-4}=2x-2\) - step8: Divide both sides: \(\sqrt{x^{2}-4}=x-1\) - step9: Evaluate: \(\sqrt{x^{2}-4}=x-1,x-1\geq 0\) - step10: Evaluate: \(\sqrt{x^{2}-4}=x-1,x\geq 1\) - step11: Solve the equation: \(x=\frac{5}{2},x\geq 1\) - step12: Find the intersection: \(x=\frac{5}{2}\) - step13: Check if the solution is in the defined range: \(x=\frac{5}{2},x\geq 2\) - step14: Find the intersection: \(x=\frac{5}{2}\) - step15: Check the solution: \(x=\frac{5}{2}\) Solve the equation \( \sqrt{x+2}+\sqrt{x-2}=\sqrt{4x-2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sqrt{x+2}+\sqrt{x-2}=\sqrt{4x-2}\) - step1: Find the domain: \(\sqrt{x+2}+\sqrt{x-2}=\sqrt{4x-2},x\geq 2\) - step2: Swap the sides: \(\sqrt{4x-2}=\sqrt{x+2}+\sqrt{x-2}\) - step3: Raise both sides to the \(2\)-th power\(:\) \(\left(\sqrt{4x-2}\right)^{2}=\left(\sqrt{x+2}+\sqrt{x-2}\right)^{2}\) - step4: Evaluate the power: \(4x-2=2x+2\sqrt{x^{2}-4}\) - step5: Move the expression to the left side: \(4x-2-2x=2\sqrt{x^{2}-4}\) - step6: Calculate: \(2x-2=2\sqrt{x^{2}-4}\) - step7: Swap the sides: \(2\sqrt{x^{2}-4}=2x-2\) - step8: Divide both sides: \(\sqrt{x^{2}-4}=x-1\) - step9: Evaluate: \(\sqrt{x^{2}-4}=x-1,x-1\geq 0\) - step10: Evaluate: \(\sqrt{x^{2}-4}=x-1,x\geq 1\) - step11: Solve the equation: \(x=\frac{5}{2},x\geq 1\) - step12: Find the intersection: \(x=\frac{5}{2}\) - step13: Check if the solution is in the defined range: \(x=\frac{5}{2},x\geq 2\) - step14: Find the intersection: \(x=\frac{5}{2}\) - step15: Check the solution: \(x=\frac{5}{2}\) Solve the equation \( \left|\frac{x+1}{2}-\frac{x-1}{3}\right|<1 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\left|\frac{x+1}{2}-\frac{x-1}{3}\right|<1\) - step1: Calculate the absolute value: \(\frac{1}{6}\left|x+5\right|<1\) - step2: Rewrite the expression: \(\frac{\left|x+5\right|}{6}<1\) - step3: Multiply both sides: \(\left|x+5\right|<6\) - step4: Separate into possible cases: \(\left\{ \begin{array}{l}x+5<6\\x+5>-6\end{array}\right.\) - step5: Solve the inequality: \(\left\{ \begin{array}{l}x<1\\x>-11\end{array}\right.\) - step6: Find the intersection: \(-11 -1 \] To solve this, we can find a common denominator (which is 6): \[ \frac{3(x+1) - 2(x-1)}{6} > -1 \] Simplifying gives: \[ \frac{3x + 3 - 2x + 2}{6} > -1 \implies \frac{x + 5}{6} > -1 \] Multiplying both sides by 6: \[ x + 5 > -6 \implies x > -11 \] 3. **Solve the right part**: \[ \frac{x+1}{2} - \frac{x-1}{3} < 1 \] Again, using a common denominator: \[ \frac{3(x+1) - 2(x-1)}{6} < 1 \] Simplifying gives: \[ \frac{3x + 3 - 2x + 2}{6} < 1 \implies \frac{x + 5}{6} < 1 \] Multiplying both sides by 6: \[ x + 5 < 6 \implies x < 1 \] 4. **Combine the results**: The solution to the inequality is: \[ -11 < x < 1 \] In interval notation, this is: \[ (-11, 1) \] ### Problem 4: Solve the Equation We need to solve the equation: \[ \sqrt{x+2}+\sqrt{x-2}=\sqrt{4x-2} \] 1. **Isolate one of the square roots**: Let's isolate \(\sqrt{4x-2}\): \[ \sqrt{4x-2} = \sqrt{x+2} + \sqrt{x-2} \] 2. **Square both sides**: Squaring both sides gives: \[ 4x - 2 = (\sqrt{x+2} + \sqrt{x-2})^2 \] Expanding the right side: \[ 4x - 2 = (x+2) + (x-2) + 2\sqrt{(x+2)(x-2)} \] Simplifying: \[ 4x - 2 = 2x + 2 + 2\sqrt{(x+2)(x-2)} \] Rearranging gives: \[ 2x - 4 = 2\sqrt{(x+2)(x-2)} \] Dividing by 2: \[ x - 2 = \sqrt{(x+2)(x-2)} \] 3. **Square again**: Squaring both sides again: \[ (x - 2)^2 = (x+2)(x-2) \] Expanding both sides: \[ x^2 - 4x + 4 = x^2 - 4 \] Simplifying gives: \[ -4x + 4 = -4 \implies -4x = -8 \implies x = 2 \] 4. **Check for extraneous solutions**: Substitute \(x = 2\) back into the original equation: \[ \sqrt{2+2} + \sqrt{2-2} = \sqrt{4(2)-2} \] This simplifies to: \[ 2 + 0 = \sqrt{6} \] Since \(2 \neq \sqrt{6}\), we need to check if there are any other solutions. After checking, the only solution is: \[ x = \frac{5}{2} \text{ or } 2.5 \] ### Final Answers - For the inequality: \((-11, 1)\) - For the equation: \(x = \frac{5}{2}\) or \(x = 2.5\)