solve the equation $\frac{1}{x + 4} - \frac{1}{x + 5} = \frac{1}{12}$.

the solutions are $x_1 =$ and $x_2 =$ where $x_1 \le x_2$.

Question

solve the equation $\frac{1}{x + 4} - \frac{1}{x + 5} = \frac{1}{12}$.

the solutions are $x_1 =$   and $x_2 =$   where $x_1 \le x_2$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Rational Equations", "Solving Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "Rational Equations", "Solving Quadratic Equations" ] } ## Combine the rational expressions Using the *Rational Equations* knowledge point $$ \begin{aligned} x & e -4, \quad x e -5 \ \frac{1}{x+4} - \frac{1}{x+5} &= \frac{(x+5) - (x+4)}{(x+4)(x+5)} \ &= \frac{1}{(x+4)(x+5)} \end{aligned} $$ ## Equate to the constant term Using the *Rational Equations* knowledge point $$ \begin{aligned} \frac{1}{(x+4)(x+5)} &= \frac{1}{12} \ (x+4)(x+5) &= 12 \end{aligned} $$ ## Expand and set to zero Using the *Solving Quadratic Equations* knowledge point $$ \begin{aligned} x^2 + 9x + 20 &= 12 \ x^2 + 9x + 8 &= 0 \end{aligned} $$ ## Solve the quadratic equation Using the *Solving Quadratic Equations* knowledge point $$ \begin{aligned} (x+1)(x+8) &= 0 \ x_1 = -8, \quad &x_2 = -1 \end{aligned} $$ Since $x_1 \le x_2$, we have $x_1 = -8$ and $x_2 = -1$. Solve the equation $\frac{1}{x+4} - \frac{1}{x+5} = \frac{1}{12}$. The solutions are $x_1 =$ -8 and $x_2 =$ -1 where $x_1 \le x_2$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Rational Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Combine the rational expressions

Equate to the constant term

Expand and set to zero

Solve the quadratic equation

Answer: