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to behaves rasional; is taking & selfi.. discusion

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Answer

x=3±10x = 3 \pm \sqrt{10}

Step 1: Identify the least common denominator (LCD).
The denominators are x−2x-2 and x+1x+1, so LCD = (x−2)(x+1)(x-2)(x+1).

Step 2: Rewrite each term with the LCD.

3(x+1)+2(x−2)(x−2)(x+1)=1\frac{3(x+1) + 2(x-2)}{(x-2)(x+1)} = 1

Step 3: Simplify the numerator.

3(x+1)+2(x−2)(x−2)(x+1)=3x+3+2x−4(x−2)(x+1)=5x−1(x−2)(x+1)=1\frac{3(x+1) + 2(x-2)}{(x-2)(x+1)} = \frac{3x + 3 + 2x - 4}{(x-2)(x+1)} = \frac{5x - 1}{(x-2)(x+1)} = 1

Step 4: Subtract 1 from both sides and simplify.

5x−1(x−2)(x+1)−1=0\frac{5x - 1}{(x-2)(x+1)} - 1 = 0 5x−1−(x−2)(x+1)(x−2)(x+1)=0\frac{5x - 1 - (x-2)(x+1)}{(x-2)(x+1)} = 0

Step 5: Expand the subtracted term.

(x−2)(x+1)=x2+x−2x−2=x2−x−2(x-2)(x+1) = x^2 + x - 2x - 2 = x^2 - x - 2

Numerator: 5x−1−(x2−x−2)=5x−1−x2+x+2=−x2+6x+15x - 1 - (x^2 - x - 2) = 5x - 1 - x^2 + x + 2 = -x^2 + 6x + 1

Step 6: Set numerator equal to zero (denominator ≠ 0).

−x2+6x+1=0⇒x2−6x−1=0-x^2 + 6x + 1 = 0 \quad \Rightarrow \quad x^2 - 6x - 1 = 0

Step 7: Solve the quadratic equation using the quadratic formula.

x=6±(−6)2−4(1)(−1)2⋅1=6±36+42=6±402=6±2102=3±10x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(-1)}}{2 \cdot 1} = \frac{6 \pm \sqrt{36 + 4}}{2} = \frac{6 \pm \sqrt{40}}{2} = \frac{6 \pm 2\sqrt{10}}{2} = 3 \pm \sqrt{10}

Step 8: Check for extraneous solutions by testing in the original equation.
For x=3+10x = 3 + \sqrt{10}: Denominators 3+10−2=1+10>03 + \sqrt{10} - 2 = 1 + \sqrt{10} > 0, 3+10+1=4+10>03 + \sqrt{10} + 1 = 4 + \sqrt{10} > 0. Valid.
For x=3−10x = 3 - \sqrt{10}: 10≈3.16\sqrt{10} \approx 3.16, so 3−3.16≈−0.16<03 - 3.16 \approx -0.16 < 0, but denominators 3−10−2=1−10<03 - \sqrt{10} - 2 = 1 - \sqrt{10} < 0, 3−10+1=4−10>03 - \sqrt{10} + 1 = 4 - \sqrt{10} > 0. Both defined and equation holds. Valid.

Final answer: x=3±10x = 3 \pm \sqrt{10}
x = 3 \pm \sqrt{10}

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Quick Answer

Identify the least common denominator (LCD). The denominators are x-2 and x+1, so LCD = (x-2)(x+1).

to behaves rasional; is taking & selfi.. discusion
Economics

This economics question tests your understanding of economic models and analysis. The step-by-step answer below applies the relevant framework and explains the reasoning.

ScanSolve AI Answer
Step 1: Identify the least common denominator (LCD). The denominators are x-2 and x+1, so LCD = (x-2)(x+1). Step 2: Rewrite each term with the LCD. (3(x+1) + 2(x-2))/((x-2)(x+1)) = 1 Step 3: Simplify the numerator. (3(x+1) + 2(x-2))/((x-2)(x+1)) = (3x + 3 + 2x - 4)/((x-2)(x+1)) = (5x - 1)/((x-2)(x+1)) = 1 Step 4: Subtract 1 from both sides and simplify. (5x - 1)/((x-2)(x+1)) - 1 = 0 (5x - 1 - (x-2)(x+1))/((x-2)(x+1)) = 0 Step 5: Expand the subtracted term. (x-2)(x+1) = x^2 + x - 2x - 2 = x^2 - x - 2 Numerator: 5x - 1 - (x^2 - x - 2) = 5x - 1 - x^2 + x + 2 = -x^2 + 6x + 1 Step 6: Set numerator equal to zero (denominator ≠ 0). -x^2 + 6x + 1 = 0 x^2 - 6x - 1 = 0 Step 7: Solve the quadratic equation using the quadratic formula. x = 6 ± sqrt((-6)^2 - 4(1)(-1))2 · 1 = 6 ± sqrt(36 + 4)2 = 6 ± sqrt(40)2 = 6 ± 2sqrt(10)2 = 3 ± sqrt(10) Step 8: Check for extraneous solutions by testing in the original equation. For x = 3 + sqrt(10): Denominators 3 + sqrt(10) - 2 = 1 + sqrt(10) > 0, 3 + sqrt(10) + 1 = 4 + sqrt(10) > 0. Valid. For x = 3 - sqrt(10): sqrt(10) ≈ 3.16, so 3 - 3.16 ≈ -0.16 < 0, but denominators 3 - sqrt(10) - 2 = 1 - sqrt(10) < 0, 3 - sqrt(10) + 1 = 4 - sqrt(10) > 0. Both defined and equation holds. Valid. Final answer: x = 3 ± sqrt(10) x = 3 ± sqrt(10)