{"id":"cb_question_bank:864f2fa0-e51c-4f6e-9d8b-178779f7b4a2","source_id":"cb_question_bank","source_item_id":"864f2fa0-e51c-4f6e-9d8b-178779f7b4a2","external_id":"176e3a8c-9f87-406f-a76a-01267378721c","ibn":null,"question_id":"5bf5136d","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.E.","skill":"Linear inequalities in one or 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":"The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 6 and 12 , which inequality represents the possible lengths, x , of the third side of the triangle?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of <math alttext=\"6\"><mn>6</mn>\n</math> and <math alttext=\"12\"><mn>12</mn>\n</math>, which inequality represents the possible lengths, <math alttext=\"x\"><mi>x</mi>\n</math>, of the third side of the triangle?</p>","rationale_html":"<p style=\"text-align: left;\">Choice C is correct. It’s given that a triangle has side lengths of <math alttext=\"6\"><mn>6</mn>\n</math> and <math alttext=\"12\"><mn>12</mn>\n</math>, and <math alttext=\"x\"><mi>x</mi>\n</math> represents the length of the third side of the triangle. It’s also given that the triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. Therefore, the inequalities <math alttext=\"6 plus x greater than 12\"><mn>6</mn><mo>+</mo><mi>x</mi><mo>&gt;</mo><mn>12</mn></math>, <math alttext=\"6 plus 12 greater than x\"><mn>6</mn><mo>+</mo><mn>12</mn><mo>&gt;</mo><mi>x</mi></math>, and <math alttext=\"12 plus x greater than 6\"><mn>12</mn><mo>+</mo><mi>x</mi><mo>&gt;</mo><mn>6</mn></math> represent all possible values of <math alttext=\"x\"><mi>x</mi>\n</math>. Subtracting <math alttext=\"6\"><mn>6</mn>\n</math> from both sides of the inequality <math alttext=\"6 plus x greater than 12\"><mn>6</mn><mo>+</mo><mi>x</mi><mo>&gt;</mo><mn>12</mn></math> yields <math alttext=\"x greater than 12 minus 6\"><mi>x</mi><mo>&gt;</mo><mn>12</mn><mo>-</mo><mn>6</mn></math>, or <math alttext=\"x greater than 6\"><mi>x</mi><mo>&gt;</mo><mn>6</mn></math>. Adding <math alttext=\"6\"><mn>6</mn>\n</math> and <math alttext=\"12\"><mn>12</mn>\n</math> in the inequality <math alttext=\"6 plus 12 greater than x\"><mn>6</mn><mo>+</mo><mn>12</mn><mo>&gt;</mo><mi>x</mi></math> yields <math alttext=\"18 greater than x\"><mn>18</mn><mo>&gt;</mo><mi>x</mi></math>, or <math alttext=\"x less than 18\"><mi>x</mi><mo>&lt;</mo><mn>18</mn></math>. Subtracting <math alttext=\"12\"><mn>12</mn>\n</math> from both sides of the inequality <math alttext=\"12 plus x greater than 6\"><mn>12</mn><mo>+</mo><mi>x</mi><mo>&gt;</mo><mn>6</mn></math> yields <math alttext=\"x greater than 6 minus 12\"><mi>x</mi><mo>&gt;</mo><mn>6</mn><mo>-</mo><mn>12</mn></math>, or <math alttext=\"x greater than negative 6\"><mi>x</mi><mo>&gt;</mo><mo>-</mo><mn>6</mn></math>. Since all <em>x</em>-values that satisfy the inequality <math alttext=\"x greater than 6\"><mi>x</mi><mo>&gt;</mo><mn>6</mn></math> also satisfy the inequality <math alttext=\"x greater than negative 6\"><mi>x</mi><mo>&gt;</mo><mo>-</mo><mn>6</mn></math>, it follows that the inequalities <math alttext=\"x greater than 6\"><mi>x</mi><mo>&gt;</mo><mn>6</mn></math> and <math alttext=\"x less than 18\"><mi>x</mi><mo>&lt;</mo><mn>18</mn></math> represent the possible values of <math alttext=\"x\"><mi>x</mi>\n</math>. Therefore, the inequality <math alttext=\"6 less than x less than 18\"><mn>6</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>18</mn></math> represents the possible lengths, <math alttext=\"x\"><mi>x</mi>\n</math>, of the third side of the triangle.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This inequality gives the upper bound for <math alttext=\"x\"><mi>x</mi>\n</math> but does not include its lower bound.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">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":"2023-08-02T20:25:59.819000+00:00","source_updated_at":"2023-08-02T20:25:59.819000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"x less than 18\"><mi>x</mi><mo>&lt;</mo><mn>18</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"x greater than 18\"><mi>x</mi><mo>&gt;</mo><mn>18</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"6 less than x less than 18\"><mn>6</mn>\n<mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>18</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"x less than 6\"><mi>x</mi><mo>&lt;</mo><mn>6</mn>\n</math> or&nbsp;<math alttext=\"x greater than 18\"><mi>x</mi><mo>&gt;</mo><mn>18</mn>\n</math></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}}