{"id":"cb_question_bank:a2183d03-c41d-442d-ac62-b13ef6b24068","source_id":"cb_question_bank","source_item_id":"a2183d03-c41d-442d-ac62-b13ef6b24068","external_id":"3855194f-20fa-4907-9dfd-8097f615bdff","ibn":null,"question_id":"fbb96bb1","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":"x minus 29 equals left parenthesis x minus a right parenthesis left parenthesis x minus 29 right parenthesis Which of the following are solutions to the given equation, where a is a constant and a greater than 30 ? a a plus 1 29","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"x minus 29 equals left parenthesis x minus a right parenthesis left parenthesis x minus 29 right parenthesis\"><mi>x</mi><mo>-</mo><mrow><mn>29</mn></mrow><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mrow><mn>29</mn></mrow></mrow></mfenced></math></p>\n<p style=\"text-align: left;\">Which of the following are solutions to the given equation, where <math alttext=\"a\"><mi>a</mi>\n</math> is a constant and <math alttext=\"a greater than 30\"><mi>a</mi><mo>&gt;</mo><mrow><mn>30</mn></mrow></math>?</p>\n<ol style=\"list-style-type: upper-roman;\">\n<li style=\"text-align: left;\"><math alttext=\"a\"><mi>a</mi>\n</math></li>\n<li style=\"text-align: left;\"><math alttext=\"a plus 1\"><mrow>\n\t<mi>a</mi>\n\t<mo>+</mo>\n\t<mn>1</mn>\n</mrow>\n</math></li>\n<li style=\"text-align: left;\"><math alttext=\"29\"><mn>29</mn>\n</math></li>\n</ol>","rationale_html":"<p>Choice C is correct. Subtracting the expression&nbsp;<math alttext=\"left parenthesis x minus 29 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced></math> from both sides of the given equation yields <math alttext=\"0 equals left parenthesis x minus a right parenthesis left parenthesis x minus 29 right parenthesis minus left parenthesis x minus 29 right parenthesis\"><mn>0</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced></math>, which can be rewritten as <math alttext=\"0 equals left parenthesis x minus a right parenthesis left parenthesis x minus 29 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis x minus 29 right parenthesis\"><mn>0</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced></math>. Since the two terms on the right-hand side of this equation have a common factor of <math alttext=\"left parenthesis x minus 29 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced></math>, it can be rewritten as <math alttext=\"0 equals left parenthesis x minus 29 right parenthesis left parenthesis x minus a plus left parenthesis negative 1 right parenthesis right parenthesis\"><mn>0</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi><mo>+</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math>, or <math alttext=\"0 equals left parenthesis x minus 29 right parenthesis left parenthesis x minus a minus 1 right parenthesis\"><mn>0</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math>. Since&nbsp;<math alttext=\"x minus a minus 1\"><mi>x</mi><mo>-</mo><mi>a</mi><mo>-</mo><mn>1</mn></math> is equivalent to <math alttext=\"x minus left parenthesis a plus 1 right parenthesis\"><mi>x</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>, the equation&nbsp;<math alttext=\"0 equals left parenthesis x minus 29 right parenthesis left parenthesis x minus a minus 1 right parenthesis\"><mn>0</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> can be rewritten as <math alttext=\"0 equals left parenthesis x minus 29 right parenthesis left parenthesis x minus left parenthesis a plus 1 right parenthesis right parenthesis\"><mn>0</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>29</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math>. By the zero product property, it follows that&nbsp;<math alttext=\"x minus 29 equals 0\"><mi>x</mi><mo>-</mo><mn>29</mn><mo>=</mo><mn>0</mn></math> or <math alttext=\"x minus left parenthesis a plus 1 right parenthesis equals 0\"><mi>x</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math>. Adding <math alttext=\"29\"><mn>29</mn>\n</math> to both sides of the equation&nbsp;<math alttext=\"x minus 29 equals 0\"><mi>x</mi><mo>-</mo><mn>29</mn><mo>=</mo><mn>0</mn></math> yields <math alttext=\"x equals 29\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>29</mn>\n</mrow>\n</math>. Adding&nbsp;<math alttext=\"a plus 1\"><mi>a</mi><mo>+</mo><mn>1</mn></math> to both sides of the equation&nbsp;<math alttext=\"x minus left parenthesis a plus 1 right parenthesis equals 0\"><mi>x</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> yields <math alttext=\"x equals a plus 1\"><mi>x</mi><mo>=</mo><mi>a</mi><mo>+</mo><mn>1</mn></math>. Therefore, the two solutions to the given equation are <math alttext=\"29\"><mn>29</mn>\n</math> and <math alttext=\"a plus 1\"><mrow>\n\t<mi>a</mi>\n\t<mo>+</mo>\n\t<mn>1</mn>\n</mrow>\n</math>. Thus, only <math alttext=\"a plus 1\"><mrow>\n\t<mi>a</mi>\n\t<mo>+</mo>\n\t<mn>1</mn>\n</mrow>\n</math> and <math alttext=\"29\"><mn>29</mn>\n</math>, not <math alttext=\"a\"><mi>a</mi>\n</math>, are solutions to the given equation.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.822000+00:00","source_updated_at":"2023-08-02T20:25:59.822000+00:00","options":[{"label":"A","content_html":"<p style=\"text-align: left;\">I and II only</p>","ord":0},{"label":"B","content_html":"<p style=\"text-align: left;\">I and III only</p>","ord":1},{"label":"C","content_html":"<p style=\"text-align: left;\">II and III only</p>","ord":2},{"label":"D","content_html":"<p style=\"text-align: left;\">I, II, and III</p>","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}