Answer
**3. Inequality Solution:**
\[
-11 < x < 1
\]
**4. Equation Solution:**
\[
x = \frac{5}{2}
\]
Solution
### 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:**
\[
x=\frac{5}{2}.
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
Explain
Simplify this solution
