{ "query": { "display": "$$\\left|x-5\\right|\\ge\\:3$$", "symbolab_question": "EQUATION#\\left|x-5\\right|\\ge 3" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Inequalities", "subTopic": "Absolute", "default": "x\\le 2\\lor x\\ge 8", "interval": "(-\\infty ,2]\\cup [8,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left|x-5\\right|\\ge\\:3{\\quad:\\quad}x\\le\\:2\\lor\\:x\\ge\\:8$$", "input": "\\left|x-5\\right|\\ge\\:3", "steps": [ { "type": "step", "primary": "Apply absolute rule: If $$|u|\\:\\geq\\:a,\\:a\\:>\\:0\\:$$then $$u\\:\\leq\\:-a\\:\\quad\\:$$or$$\\quad\\:u\\:\\geq\\:a$$", "result": "x-5\\le\\:-3\\lor\\:x-5\\ge\\:3" }, { "type": "interim", "title": "$$x-5\\le\\:-3\\lor\\:x-5\\ge\\:3$$", "input": "x-5\\le\\:-3\\lor\\:x-5\\ge\\:3", "result": "x\\le\\:2\\lor\\:x\\ge\\:8", "steps": [ { "type": "interim", "title": "$$x-5\\le\\:-3{\\quad:\\quad}x\\le\\:2$$", "input": "x-5\\le\\:-3", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "x-5\\le\\:-3", "result": "x\\le\\:2", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "x-5+5\\le\\:-3+5" }, { "type": "step", "primary": "Simplify", "result": "x\\le\\:2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-5\\ge\\:3{\\quad:\\quad}x\\ge\\:8$$", "input": "x-5\\ge\\:3", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "x-5\\ge\\:3", "result": "x\\ge\\:8", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "x-5+5\\ge\\:3+5" }, { "type": "step", "primary": "Simplify", "result": "x\\ge\\:8" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "step", "primary": "Combine the intervals", "result": "x\\le\\:2\\lor\\:x\\ge\\:8" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Inequalities" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "x=2", "displayFormula": "x=2", "attributes": { "color": "GRAY", "lineType": "BOLD", "isAsymptote": false } }, { "evalFormula": "x=8", "displayFormula": "x=8", "attributes": { "color": "GRAY", "lineType": "BOLD", "isAsymptote": false } } ] }, "fills": [ { "ranges": [ { "fst": "-Infinity", "snd": "2.0" }, { "fst": "8.0", "snd": "Infinity" } ], "funcIndices": [], "funcs": [], "xIneq": true, "yIneq": false, "twoVar": false, "trueAboveLine": false, "color": "rgba(171, 181, 235, 0.3)" } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "x\\le 2\\lor x\\ge 8", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -1.5714285714285714, "xMax": 10, "yMin": -2, "yMax": 8 }, "intervals": [ { "maxd": 2, "minInclusive": false, "maxInclusive": true, "maxFinite": true, "minFinite": false, "isAllR": false, "isPoint": false }, { "mind": 8, "minInclusive": true, "maxInclusive": false, "maxFinite": false, "minFinite": true, "isAllR": false, "isPoint": false } ], "imageAPI": "x\\le 2\\lor x\\ge 8" }, "showViewLarger": true } }, "meta": { "showVerify": true } }