20. Determine at least one real solution for x that satisfies the equation shown below: (2x)/(x+3)-3/(x+1)=-1

1. Eliminate Fractions<br /> Multiply both sides by $(x+3)(x+1)$ to clear the fractions: $2x(x+1) - 3(x+3) = -(x+3)(x+1)$.<br /><br />2. Expand and Simplify<br /> Expand: $2x^2 + 2x - 3x - 9 = -x^2 - 4x - 3$.<br /><br />3. Combine Like Terms<br /> Combine terms: $2x^2 + 5x + 9 = -x^2 - 4x - 3$.<br /><br />4. Move All Terms to One Side<br /> Rearrange: $2x^2 + x^2 + 5x + 4x + 9 + 3 = 0$, resulting in $3x^2 + 9x + 12 = 0$.<br /><br />5. Factor the Quadratic Equation<br /> Factor: $3(x^2 + 3x + 4) = 0$. The quadratic $x^2 + 3x + 4$ does not factor easily, so use the quadratic formula.<br /><br />6. Apply the Quadratic Formula<br /> Use $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=3$, $c=4$: $x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$.<br /><br />7. Calculate the Discriminant<br /> Calculate: $3^2 - 16 = 9 - 16 = -7$. Since the discriminant is negative, there are no real solutions.