To solve the equation [tex]\(4 + \frac{3}{x - 2} = \frac{3}{x - 3}\)[/tex], let's follow a step-by-step process.
1. Move the constants and fractions to one side:
[tex]\[4 + \frac{3}{x - 2} = \frac{3}{x - 3}\][/tex]
2. Isolate the term with fractions:
[tex]\[\frac{3}{x - 2} - \frac{3}{x - 3} = -4\][/tex]
3. Factor out the common factor [tex]\(3\)[/tex] in the numerators:
[tex]\[3\left(\frac{1}{x - 2} - \frac{1}{x - 3}\right) = -4\][/tex]
4. Combine the fractions on the left-hand side:
[tex]\[3 \left( \frac{(x - 3) - (x - 2)}{(x - 2)(x - 3)} \right) = -4\][/tex]
Simplify the numerator:
[tex]\[3 \left( \frac{x - 3 - x + 2}{(x - 2)(x - 3)} \right) = -4\][/tex]
[tex]\[3 \left( \frac{-1}{(x - 2)(x - 3)} \right) = -4\][/tex]
5. Simplify the expression:
[tex]\[\frac{-3}{(x - 2)(x - 3)} = -4\][/tex]
6. Eliminate the negative signs by multiplying both sides by [tex]\(-1\)[/tex]:
[tex]\[\frac{3}{(x - 2)(x - 3)} = 4\][/tex]
7. Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[3 = 4(x - 2)(x - 3)\][/tex]
8. Divide both sides by 4 to isolate [tex]\(x\)[/tex]:
[tex]\[\frac{3}{4} = (x - 2)(x - 3)\][/tex]
9. Expand the product on the right side:
[tex]\[\frac{3}{4} = x^2 - 5x + 6\][/tex]
10. Multiply both sides by 4 to clear the fraction:
[tex]\[3 = 4x^2 - 20x + 24\][/tex]
11. Rearrange the equation to form a standard quadratic equation:
[tex]\[4x^2 - 20x + 24 - 3 = 0\][/tex]
[tex]\[4x^2 - 20x + 21 = 0\][/tex]
12. Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 4\)[/tex], [tex]\(b = -20\)[/tex], and [tex]\(c = 21\)[/tex].
Calculate the discriminant:
[tex]\[b^2 - 4ac = (-20)^2 - 4(4)(21)\][/tex]
[tex]\[= 400 - 336\][/tex]
[tex]\[= 64\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[x = \frac{-(-20) \pm \sqrt{64}}{2 \cdot 4}\][/tex]
[tex]\[x = \frac{20 \pm 8}{8}\][/tex]
Find the two solutions:
[tex]\[x = \frac{20 + 8}{8} = \frac{28}{8} = \frac{7}{2}\][/tex]
[tex]\[x = \frac{20 - 8}{8} = \frac{12}{8} = \frac{3}{2}\][/tex]
Therefore, the solutions to the equation [tex]\(4 + \frac{3}{x - 2} = \frac{3}{x - 3}\)[/tex] are [tex]\(x = \frac{3}{2}, \frac{7}{2}\)[/tex].
