domain of (7x)/(5+3x)

{ "query": { "display": "domain $$\\frac{7x}{5+3x}$$", "symbolab_question": "FUNCTION#domain \\frac{7x}{5+3x}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "domain", "default": "x<-\\frac{5}{3}\\lor x>-\\frac{5}{3}", "interval": "(-\\infty ,-\\frac{5}{3})\\cup (-\\frac{5}{3},\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Domain of $$\\frac{7x}{5+3x}\\::{\\quad}x<-\\frac{5}{3}\\lor\\:x>-\\frac{5}{3}$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=-\\frac{5}{3}$$", "input": "\\frac{7x}{5+3x}", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{7x}{5+3x}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$5+3x=0:{\\quad}x=-\\frac{5}{3}$$", "input": "5+3x=0", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "5+3x=0", "result": "3x=-5", "steps": [ { "type": "step", "primary": "Subtract $$5$$ from both sides", "result": "5+3x-5=0-5" }, { "type": "step", "primary": "Simplify", "result": "3x=-5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=-5", "result": "x=-\\frac{5}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{-5}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=-\\frac{5}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=-\\frac{5}{3}" } ], "meta": { "interimType": "Undefined Points 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzmeFdjgvhQTm0mwfI9L6ocFLSGAzzjjy2VnK9x0onyXkroQM2YgsjGnOiDXPf01nBFeWrgRJMXuH/85lf1j/JKdriiCv9ZiECl6FT7v91CPfxi6PFTq1Sy+8pWTD/50TeQKHeh69S6dnv9vSoUoFKObI2tXvUGQYT0L6wdpcOiJGB1AkhCJy5xKRnb/nUvuTJPvfJZfrYPC3p72Ab9Sgg==" } }, { "type": "step", "primary": "The function domain", "result": "x<-\\frac{5}{3}\\lor\\:x>-\\frac{5}{3}" } ], "meta": { "solvingClass": "Function Domain" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{7x}{5+3x}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }