{"id":"cb_question_bank:3e8dfd58-78d1-40c2-a518-ef7bf4964012","source_id":"cb_question_bank","source_item_id":"3e8dfd58-78d1-40c2-a518-ef7bf4964012","external_id":"57e960ad-dbb1-486d-9f07-1da462789c42","ibn":null,"question_id":"541bef2f","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":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"y less than or equals x plus 7 y greater than or equals negative 2 x minus 1 Which point left parenthesis x comma y right parenthesis is a solution to the given system of inequalities in the xy -plane?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"y less than or equals x plus 7\"><mi>y</mi><mo>≤</mo><mi>x</mi><mo>+</mo><mrow><mn>7</mn></mrow></math></p>\n<p style=\"text-align: center;\"><math alttext=\"y greater than or equals negative 2 x minus 1\"><mi>y</mi><mo>≥</mo><mrow><mo>-</mo><mn>2</mn></mrow><mi>x</mi><mo>-</mo><mrow><mn>1</mn></mrow></math></p>\n<p>Which point <math alttext=\"left parenthesis x comma y right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> is a solution to the given system of inequalities in the <em>xy</em>-plane?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. A point&nbsp;<math alttext=\"left parenthesis x comma y right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> is a solution to a system of inequalities in the <em>xy</em>-plane if substituting the <em>x</em>-coordinate and the <em>y</em>-coordinate of the point for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>, respectively, in each inequality makes both of the inequalities true. Substituting the <em>x</em>-coordinate and the <em>y</em>-coordinate of choice D, <math alttext=\"14\"><mn>14</mn>\n</math> and <math alttext=\"0\"><mn>0</mn>\n</math>, for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"y\"><mi>y</mi>\n</math>, respectively, in the first inequality in the given system, <math alttext=\"y less than or equals x plus 7\"><mi>y</mi><mo>≤</mo><mi>x</mi><mo>+</mo><mn>7</mn></math>, yields&nbsp;<math alttext=\"0 less than or equals 14 plus 7\"><mn>0</mn><mo>≤</mo><mn>14</mn><mo>+</mo><mn>7</mn></math>, or&nbsp;<math alttext=\"0 less than or equals 21\"><mn>0</mn><mo>≤</mo><mn>21</mn></math>, which is true. Substituting <math alttext=\"14\"><mn>14</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the second inequality in the given system, <math alttext=\"y greater than or equals minus 2 x minus 1\"><mi>y</mi><mo>≥</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></math>, yields&nbsp;<math alttext=\"0 greater than or equals minus 2 left parenthesis 14 right parenthesis minus 1\"><mn>0</mn><mo>≥</mo><mo>-</mo><mn>2</mn><mfenced><mn>14</mn></mfenced><mo>-</mo><mn>1</mn></math>, or&nbsp;<math alttext=\"0 greater than or equals negative 29\"><mn>0</mn><mo>≥</mo><mo>-</mo><mn>29</mn></math>, which is true. Therefore, the point&nbsp;<math alttext=\"left parenthesis 14 comma 0 right parenthesis\"><mfenced><mrow><mn>14</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> is a solution to the given system of inequalities in the <em>xy</em>-plane.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. Substituting <math alttext=\"negative 14\"><mo>-</mo><mn>14</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the inequality <math alttext=\"y less than or equals x plus 7\"><mi>y</mi><mo>≤</mo><mi>x</mi><mo>+</mo><mn>7</mn></math>&nbsp;yields&nbsp;<math alttext=\"0 less than or equals negative 14 plus 7\"><mn>0</mn><mo>≤</mo><mo>-</mo><mn>14</mn><mo>+</mo><mn>7</mn></math>, or&nbsp;<math alttext=\"0 less than or equals negative 7\"><mn>0</mn><mo>≤</mo><mo>-</mo><mn>7</mn></math>, which is not true.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"negative 14\"><mo>-</mo><mn>14</mn>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the inequality&nbsp;<math alttext=\"y greater than or equals minus 2 x minus 1\"><mi>y</mi><mo>≥</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></math> yields&nbsp;<math alttext=\"negative 14 greater than or equals minus 2 left parenthesis 0 right parenthesis minus 1\"><mo>-</mo><mn>14</mn><mo>≥</mo><mo>-</mo><mn>2</mn><mfenced><mn>0</mn></mfenced><mo>-</mo><mn>1</mn></math>, or&nbsp;<math alttext=\"negative 14 greater than or equals negative 1\"><mo>-</mo><mn>14</mn><mo>≥</mo><mo>-</mo><mn>1</mn></math>, which is not true.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"14\"><mn>14</mn>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the inequality&nbsp;<math alttext=\"y less than or equals x plus 7\"><mi>y</mi><mo>≤</mo><mi>x</mi><mo>+</mo><mn>7</mn></math> yields&nbsp;<math alttext=\"14 less than or equals 0 plus 7\"><mn>14</mn><mo>≤</mo><mn>0</mn><mo>+</mo><mn>7</mn></math>, or&nbsp;<math alttext=\"14 less than or equals 7\"><mn>14</mn><mo>≤</mo><mn>7</mn></math>, which is not true.</p>","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.820000+00:00","source_updated_at":"2023-08-02T20:25:59.820000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"left parenthesis negative 14 comma 0 right parenthesis\"><mfenced><mrow><mrow><mo>-</mo><mn>14</mn></mrow><mo>,</mo><mn>0</mn></mrow></mfenced></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"left parenthesis 0 comma negative 14 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>14</mn></mrow></mrow></mfenced></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"left parenthesis 0 comma 14 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mrow><mn>14</mn></mrow></mrow></mfenced></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"left parenthesis 14 comma 0 right parenthesis\"><mfenced><mrow><mrow><mn>14</mn></mrow><mo>,</mo><mn>0</mn></mrow></mfenced></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}}