{"id":"cb_question_bank:e9c7d460-326e-4a56-9d1b-700b343e12f0","source_id":"cb_question_bank","source_item_id":"e9c7d460-326e-4a56-9d1b-700b343e12f0","external_id":"df05bbf9-4751-4ac6-b485-8df58c021387","ibn":null,"question_id":"da602115","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":"M","score_band":4,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"If StartAbsoluteValue 4 x minus 4 EndAbsoluteValue equals 112 , what is the positive value of x minus 1 ?","stimulus":"","stimulus_html":null,"stem_html":"<p>If <math alttext=\"StartAbsoluteValue 4 x minus 4 EndAbsoluteValue equals 112\"><mfenced><mrow><mrow><mn>4</mn></mrow><mi>x</mi><mo>-</mo><mrow><mn>4</mn></mrow></mrow></mfenced><mo>=</mo><mrow><mn>112</mn></mrow></math>, what is the positive value of <math alttext=\"x minus 1\"><mrow>\n\t<mi>x</mi>\n\t<mo>-</mo>\n\t<mn>1</mn>\n</mrow>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"28\"><mn>28</mn>\n</math>. The given absolute value equation can be rewritten as two linear equations: <math alttext=\"4 x minus 4 equals 112\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>4</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>4</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>112</mn>\n</mrow>\n</math> and <math alttext=\"minus left parenthesis 4 x minus 4 right parenthesis equals 112\"><mo>-</mo><mfenced><mrow><mn>4</mn><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>112</mn></math>, or <math alttext=\"4 x minus 4 equals negative 112\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>4</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>4</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>112</mn>\n</mrow>\n</math>. Adding <math alttext=\"4\"><mn>4</mn>\n</math> to both sides of the equation <math alttext=\"4 x minus 4 equals 112\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>4</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>4</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>112</mn>\n</mrow>\n</math> results in <math alttext=\"4 x equals 116\"><mrow>\n\t<mrow>\n\t\t<mn>4</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>116</mn>\n</mrow>\n</math>. Dividing both sides of this equation by <math alttext=\"4\"><mn>4</mn>\n</math> results in <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 <math alttext=\"4\"><mn>4</mn>\n</math> to both sides of the equation <math alttext=\"4 x minus 4 equals negative 112\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>4</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>4</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>112</mn>\n</mrow>\n</math> results in <math alttext=\"4 x equals negative 108\"><mrow>\n\t<mrow>\n\t\t<mn>4</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>108</mn>\n</mrow>\n</math>. Dividing both sides of this equation by <math alttext=\"4\"><mn>4</mn>\n</math> results in <math alttext=\"x equals negative 27\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>27</mn>\n</mrow>\n</math>. Therefore, the two values of <math alttext=\"x minus 1\"><mrow>\n\t<mi>x</mi>\n\t<mo>-</mo>\n\t<mn>1</mn>\n</mrow>\n</math> are <math alttext=\"29 minus 1\"><mn>29</mn><mo>-</mo><mn>1</mn></math>, or <math alttext=\"28\"><mn>28</mn>\n</math>, and <math alttext=\"negative 27 minus 1\"><mo>-</mo><mn>27</mn><mo>-</mo><mn>1</mn></math>, or <math alttext=\"negative 28\"><mo>-</mo><mn>28</mn>\n</math>. Thus, the positive value of <math alttext=\"x minus 1\"><mrow>\n\t<mi>x</mi>\n\t<mo>-</mo>\n\t<mn>1</mn>\n</mrow>\n</math> is <math alttext=\"28\"><mn>28</mn>\n</math>.</p>\n<p style=\"text-align: left;\">Alternate approach: The given equation can be rewritten as&nbsp;<math alttext=\"StartAbsoluteValue 4 left parenthesis x minus 1 right parenthesis EndAbsoluteValue equals 112\"><mfenced><mrow><mn>4</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>112</mn></math>, which is equivalent to&nbsp;<math alttext=\"4 StartAbsoluteValue x minus 1 EndAbsoluteValue equals 112\"><mn>4</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>112</mn></math>. Dividing both sides of this equation by <math alttext=\"4\"><mn>4</mn>\n</math> yields&nbsp;<math alttext=\"StartAbsoluteValue x minus 1 EndAbsoluteValue equals 28\"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>28</mn></math>. This equation can be rewritten as two linear equations: <math alttext=\"x minus 1 equals 28\"><mrow>\n\t<mrow>\n\t\t<mi>x</mi>\n\t\t<mo>-</mo>\n\t\t<mn>1</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>28</mn>\n</mrow>\n</math>&nbsp;and&nbsp;<math alttext=\"minus left parenthesis x minus 1 right parenthesis equals 28\"><mo>-</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>28</mn></math>, or <math alttext=\"x minus 1 equals negative 28\"><mrow>\n\t<mrow>\n\t\t<mi>x</mi>\n\t\t<mo>-</mo>\n\t\t<mn>1</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>28</mn>\n</mrow>\n</math>. Therefore, the positive value of <math alttext=\"x minus 1\"><mrow>\n\t<mi>x</mi>\n\t<mo>-</mo>\n\t<mn>1</mn>\n</mrow>\n</math> is <math alttext=\"28\"><mn>28</mn>\n</math>.</p>","correct_answer":["28"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.823000+00:00","source_updated_at":"2023-08-02T20:25:59.823000+00:00","options":[],"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}}