{"id":"cb_question_bank:cd688f08-98d4-42aa-a3e5-1049b6f47d49","source_id":"cb_question_bank","source_item_id":"cd688f08-98d4-42aa-a3e5-1049b6f47d49","external_id":null,"ibn":"025616-DC","question_id":"fc3dfa26","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.B.","skill":"Nonlinear equations in one variable and systems of equations in two variables","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"What value of x satisfies the equation above?","stimulus":"the fraction with numerator 4 x squared, and denominator x squared minus 9, end fraction, minus the fraction with numerator 2 x, and denominator x plus 3, end fraction, equals the fraction with numerator 1, and denominator x minus 3, end fraction","stimulus_html":"<div class=\"stimulus_reference \">\n        <p class=\"standalone_statement style:1 \">\n          <span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9320fccc2201239e6b3e737cb8a4dfe8986e0acddc46eccfcd5292aff63b3a9d\" alt=\"the fraction with numerator 4 x squared, and denominator x squared minus 9, end fraction, minus the fraction with numerator 2 x, and denominator x plus 3, end fraction, equals the fraction with numerator 1, and denominator x minus 3, end fraction\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">What value of <span class=\"italic\">x</span> satisfies the equation above?</p>\n","rationale_html":"<p>Choice C is correct. Each fraction in the given equation can be expressed with the common denominator <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4dacacaeacc4660d6824bc5426d3becb4493dc2d4b5857b104c5d1a2c94f15eb\" alt=\"x squared, minus 9\"></span>. Multiplying <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1ea57cf15c4843d8d187493cf91894716ac928a9f3faac8b9764d0ac25334b99\" alt=\"the fraction with numerator 2 x, and denominator x plus 3, end fraction\"></span> by <sup><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3f23fed630b11374a705f16bacfdb0b47c6c47b8933f4778d94e6e01bb57b6bc\" alt=\"the fraction with numerator x minus 3, and denominator x minus 3, end fraction \"></span></sup> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0e11b23e22ae395f3104a2ee5d897e5579a001663ef0170a38bae0b21e149c01\" alt=\"the fraction with numerator 2 x squared, minus 6, and denominator x squared minus 9, end fraction \"></span>, and multiplying <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fc6889a7e8059fc431374c6688cdb261a4c83b8d403e3dc61c6191bc69aec69f\" alt=\"the fraction with numerator 1, and denominator x minus 3, end fraction \"></span> by <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ce3a36ee3b9a075627d8e12d69099c2c605e3fcbe186d8b8317cbacb5857412d\" alt=\"the fraction with numerator x plus 3, and denominator x plus 3, end fraction\"></span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a2ebc545b5cc73141c2605b1e470e29bc08fe4005f0947d6fd91afd933d33610\" alt=\"the fraction with numerator x plus 3, and denominator x squared minus 9, end fraction\"></span>. Therefore, the given equation can be written as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0effb7750ce8e4cdc6d077dc721b2cefc31045db12164e9805b884c2981e7b63\" alt=\"the fraction with numerator 4 x squared, and denominator x squared minus 9, minus, the fraction with numerator 2 x squared, minus 6 x, and denominator x squared minus 9, end fraction, equals, the fraction with numerator x plus 3, and denominator x squared minus 9, end fraction\"></span>. Multiplying each fraction by the denominator results in the equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bac01dd50ad30cfd8b1efd2dee06b1a1aa7530563c80f967c531b3c87f6e051f\" alt=\"4 x squared minus, open parenthesis, 2 x squared, minus 6 x, close parenthesis, equals, x plus 3\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cd26e78ee1f8c66c03923ad942fee24bd87bd3c817648c0c373acabe9d3dbba5\" alt=\"2 x squared, plus 6 x, equals, x plus 3\"></span>. This equation can be solved by setting a quadratic expression equal to 0, then solving for <span class=\"italic\">x</span>. Subtracting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/210c4235c5d6f2007ee90909378c0dd9b57ac9dc70f0cb76918e8ce80a70cc40\" alt=\"x plus 3 \"></span> from both sides of this equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c14ff4390070ddfd00c07899c63023dc79de5ca28d370d6b198128653c16e595\" alt=\"2 x squared, plus 5 x, minus 3, equals 0\"></span>. The expression <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/428f004afd415f2a32cda5b1d4c5df78e0d1ec12c34a0380265a405fa7ae50f4\" alt=\"2 x squared, plus 5 x, minus 3 \"></span> can be factored, resulting in the equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/11576b6c74a0bd0f6c0105db5447b1ed7fda7896525d6f89fe1cef5721ee40ff\" alt=\"open parenthesis, 2 x minus 1, close parenthesis, times, open parenthesis, x plus 3, close parenthesis, equals 0\"></span>. By the zero product property, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/200840a486e2d780469d040f34a80255a4db3f555b138d40ad883fa3e12e7ccf\" alt=\"2 x minus 1, equals 0\"></span> or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fddf943de833092e49d4dae37826f1a121f2fe5f1cfb511fb259e1f858a70dc6\" alt=\"x plus 3, equals 0\"></span>. To solve for <span class=\"italic\">x</span> in <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/200840a486e2d780469d040f34a80255a4db3f555b138d40ad883fa3e12e7ccf\" alt=\"2 x minus 1, equals 0\"></span>, 1 can be added to both sides of the equation, resulting in <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9e6e743dd0eae20cca8f183ce93e1161d74262a767043aaab00fba0429741886\" alt=\"2 x equals 1\"></span>. Dividing both sides of this equation by 2 results in <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/da05b2646b35810c7b8c04bb41de46f0269aaa835008c9a56aeed0fb0edc1bc5\" alt=\"x equals one half\"></span>. Solving for <span class=\"italic\">x </span>in <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fddf943de833092e49d4dae37826f1a121f2fe5f1cfb511fb259e1f858a70dc6\" alt=\"x plus 3, equals 0\"></span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8e5f912b3fb6f3c9039c512de1942fa4f3d22df1ab62e351912ce1c61fb822e2\" alt=\"x equals negative 3\"></span>. However, this value of <span class=\"italic\">x</span> would result in the second fraction of the original equation having a denominator of 0. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8e5f912b3fb6f3c9039c512de1942fa4f3d22df1ab62e351912ce1c61fb822e2\" alt=\"x equals negative 3\"></span> is an extraneous solution. Thus, the only value of <span class=\"italic\">x</span> that satisfies the given equation is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/da05b2646b35810c7b8c04bb41de46f0269aaa835008c9a56aeed0fb0edc1bc5\" alt=\"x equals one half\"></span>.</p><p>Choice A is incorrect and may result from solving <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fddf943de833092e49d4dae37826f1a121f2fe5f1cfb511fb259e1f858a70dc6\" alt=\"x plus 3, equals 0 \"></span> 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 <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/200840a486e2d780469d040f34a80255a4db3f555b138d40ad883fa3e12e7ccf\" alt=\"2 x minus 1, equals 0 \"></span> for <span class=\"italic\">x</span>. Choice D is incorrect and may result from a calculation error.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.619000+00:00","source_updated_at":"2023-08-02T20:25:59.619000+00:00","options":[{"label":"A","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fcb3b618cff214ac5a09c033eaa100f5ee4fc55b088acb71912333adf75e3f91\" alt=\"negative 3\"></span></span>\n          </span>\n","ord":0},{"label":"B","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c32675bf7ded3a05413c4197ed13d278dd4cd18b26f7e63428e41d3fb6ed70bb\" alt=\"negative one half\"></span></span>\n          </span>\n","ord":1},{"label":"C","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/acd304db4db3ee4a427462753769f86b9941b320104a3856da731c39db5d894c\" alt=\"one half\"></span></span>\n          </span>\n","ord":2},{"label":"D","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b35ae18904fb781bd47e663cf02b992965491e4f50b31194e7d69870143879cd\" alt=\"3\"></span>\n","ord":3}],"attribution":null}