{"@context":"https://schema.org/","@type":"Quiz","name":"

$\\frac{4x^{2}}{x^{2}-9}-\\frac{2x}{x+3}=\\frac{1}{x-3}$

What value of $x$ satisfies the equation above?

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-174"}],"about":{"@type":"Thing","name":"Math, Advanced Math, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Advanced Math","Duplicate"],"name":"

$\\frac{4x^{2}}{x^{2}-9}-\\frac{2x}{x+3}=\\frac{1}{x-3}$

What value of $x$ satisfies the equation above?

","comment":{"@type":"Comment","text":"

Choice C is correct. Each fraction in the given equation can be expressed with the common denominator $x^{2}-9$.

Multiplying $\\frac{2x}{x+3}$ by $\\frac{x-3}{x-3}$ yields $\\frac{2x^{2}-6x}{x^{2}-9}$, and multiplying $\\frac{1}{x-3}$ by $\\frac{x+3}{x+3}$ yields $\\frac{x+3}{x^{2}-9}$. Therefore, the given equation can be written as $\\frac{4x^{2}}{x^{2}-9}-\\frac{2x^{2}-6x}{x^{2}-9}=\\frac{x+3}{x^{2}-9}$. Multiplying each fraction by the denominator results in the equation $4x^{2}-(2x^{2}-6x)=x+3$, or $2x^{2}+6x=x+3$. This equation can be solved by setting a quadratic expression equal to 0, then solving for $x$. Subtracting $x+3$ from both sides of this equation yields $2x^{2}+5x-3=0$. The expression $2x^{2}+5x-3$ can be factored, resulting in the equation $(2x-1)(x+3)=0$. By the zero product property, $2x-1=0$ or $x+3=0$. To solve for $x$ in $2x-1=0$, 1 can be added to both sides of the equation, resulting in $2x=1$. Dividing both sides of this equation by 2 results in $x=\\frac{1}{2}$. Solving for $x$ in $x+3=0$ yields $x=-3$. However, this value of $x$ would result in the second fraction of the original equation having a denominator of 0. Therefore, $x=-3$ is an extraneous solution. Thus, the only value of x that satisfies the given equation is $x=\\frac{1}{2}$.

Choice A is incorrect and may result from solving $x+3=0$ but not realizing that this solution is extraneous because it would result in a denominator of 0 in the second fraction. Choice B is incorrect and may result from a sign error when solving $2x-1=0$ for $x$. Choice D is incorrect and may result from a calculation error.

"},"encodingFormat":"text/markdown","text":"

$\\frac{4x^{2}}{x^{2}-9}-\\frac{2x}{x+3}=\\frac{1}{x-3}$

What value of $x$ satisfies the equation above?

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$-3$

"},{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$-\\frac{1}{2}$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$3$

"}],"acceptedAnswer":{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$\\frac{1}{2}$

","comment":{"@type":"Comment","text":"

"},"answerExplanation":{"@type":"Comment","text":"

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-advanced-math-multiple-choice-question-174"}

$\frac{4x^{2}}{x^{2}-9}-\frac{2x}{x+3}=\frac{1}{x-3}$

What value of $x$ satisfies the equation above?