how many extraneous solutions does the equation below have?
$\frac{3m}{2m+3} - \frac{2m}{2m-3} = 1$

* 0
* 1
* 2
* 3

Question

how many extraneous solutions does the equation below have?
$\frac{3m}{2m+3} - \frac{2m}{2m-3} = 1$

* 0
* 1
* 2
* 3

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Solving Rational Equations", "Extraneous Solutions" ], "new_concepts": [], "current_concepts": [ "Solving Rational Equations", "Extraneous Solutions" ] } ## Identify excluded values Using the **Solving Rational Equations** and **Extraneous Solutions** knowledge points $$ \begin{aligned} &2m + 3 e 0 \implies m e -\frac{3}{2}\ &2m - 3 e 0 \implies m e \frac{3}{2} \end{aligned} $$ ## Clear the denominators Using the **Solving Rational Equations** knowledge point $$ \begin{aligned} &\frac{3m}{2m+3} - \frac{2m}{2m-3} = 1\ &3m(2m-3) - 2m(2m+3) = (2m+3)(2m-3)\ &6m^2 - 9m - (4m^2 + 6m) = 4m^2 - 9\ &2m^2 - 15m = 4m^2 - 9 \end{aligned} $$ ## Solve the quadratic equation Using the **Solving Rational Equations** knowledge point $$ \begin{aligned} &2m^2 + 15m - 9 = 0\ &m = \frac{-15 \pm \sqrt{15^2 - 4(2)(-9)}}{2(2)}\ &m = \frac{-15 \pm \sqrt{225 + 72}}{4} = \frac{-15 \pm \sqrt{297}}{4} \end{aligned} $$ ## Check for extraneous solutions Using the **Extraneous Solutions** knowledge point $$ \begin{aligned} &m = \frac{-15 \pm 3\sqrt{33}}{4}\ &m e \pm\frac{3}{2} \implies \text{No solutions are extraneous.} \end{aligned} $$

(A) 0

(B) 1

(C) 2

(D) 3

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Solving Rational Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Identify excluded values

Clear the denominators

Solve the quadratic equation

Check for extraneous solutions

Answer: