(2x-5)/((3x+5)(x-8)(x-2))<0

{ "query": { "display": "$$\\frac{2x-5}{\\left(3x+5\\right)\\left(x-8\\right)\\left(x-2\\right)}<0$$", "symbolab_question": "EQUATION#\\frac{2x-5}{(3x+5)(x-8)(x-2)}<0" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Inequalities", "subTopic": "RationalIneqSolver", "default": "-\\frac{5}{3}<x<2\\lor \\frac{5}{2}<x<8", "interval": "(-\\frac{5}{3},2)\\cup (\\frac{5}{2},8)", "decimal": "-1.66666…<x<2\\lor 2.5<x<8", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{2x-5}{\\left(3x+5\\right)\\left(x-8\\right)\\left(x-2\\right)}<0{\\quad:\\quad}-\\frac{5}{3}<x<2\\lor\\:\\frac{5}{2}<x<8$$", "input": "\\frac{2x-5}{\\left(3x+5\\right)\\left(x-8\\right)\\left(x-2\\right)}<0", "steps": [ { "type": "interim", "title": "Identify the intervals", "result": "-\\frac{5}{3}<x<2\\lor\\:\\frac{5}{2}<x<8", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\frac{2x-5}{\\left(3x+5\\right)\\left(x-8\\right)\\left(x-2\\right)}$$" }, { "type": "interim", "title": "Find the signs of $$2x-5$$", "steps": [ { "type": "interim", "title": "$$2x-5=0:{\\quad}x=\\frac{5}{2}$$", "input": "2x-5=0", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "2x-5=0", "result": "2x=5", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "2x-5+5=0+5" }, { "type": "step", "primary": "Simplify", "result": "2x=5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vtP5cQK1DOePqRSAUs12vxyOUtErWNRXbTqqjj69E3v0/0aIMjdGoc6L+9BkyFeXRKvwpCsMm/dIDWKdnSUJtGCLp1O9ogni8gQnTAGnqpGTDp38YGC88uSLUJ2/3TjNBII9y7a9iu65XQd3qNxlcHOuJioD0NRqN1Eah09L4f+/AqjyyjhUTD50HesEa0Dw7qz0Y/8NSmMUV1FULviXVKU5YxDLzW4dJ11H1VcrdkO0liQlMRTzJUmAIbvvxAK5bgJv9ltOmEssIBRWD1cpqd8dD0Lo6CIUNEYCqtsfAHgm/U1KHQsVoHy2nFpFSDpBjjVSpBZkaPfgpLmaKJuiP3x+pFb2rvhDUe2sIQFPphISlu7T2Vr3K5EGofr4zvhrz9+gNlt8jys/z5Nqw8FxnRbe2Wyzan9NYP6IQI2E3rk=" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=5", "result": "x=\\frac{5}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{5}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{5}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7P3U+Yq35py9FkaUtloBi6nqmU5rQtJ8zdtqJ0kjhpCmAihI8YpOa7YjcZorvECwsGTs6/dv6vqGrZei/KKb0d7fMx6nqIiE41EU3nq3E9Q8KA/ZpOZ/69yR+tuSn6BgZ0hYzF6vxjSZoMSInCY/PPQbO95/2yiZOrh8Vdx/OHPtolA7ljGwKjl4uJ3NTxfZc0+cMFhWhkyzb1aDn8WqP9U4LVfFHk5To268IqjZ58KLSYaQPf7g/ckfz4eqmlhX1OSsh7K0N7TukStXuv1XfrJWtRhIg+0KEyyHjCzeI3B/QaipHbr1nmdWpXYzbn83Bys4JCfXqVwFRIPJNMoUBh3rYYacq0E3WuEr1HGZvyrCREJa0Ou6rrvOkQ7oxS7UugIoSPGKTmu2I3GaK7xAsLDbcpGIV7EJ/S1nHIdsy+v0=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$2x-5<0:{\\quad}x<\\frac{5}{2}$$", "input": "2x-5<0", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "2x-5<0", "result": "2x<5", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "2x-5+5<0+5" }, { "type": "step", "primary": "Simplify", "result": "2x<5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gohIhRyz6GwxjFmyyiaBWxyOUtErWNRXbTqqjj69E3v0/0aIMjdGoc6L+9BkyFeXRKvwpCsMm/dIDWKdnSUJtE/7ZstdQ6WO7sk8VgRGeV2TDp38YGC88uSLUJ2/3TjNBII9y7a9iu65XQd3qNxlcHOuJioD0NRqN1Eah09L4f+/AqjyyjhUTD50HesEa0Dw7qz0Y/8NSmMUV1FULviXVKU5YxDLzW4dJ11H1VcrdkO0liQlMRTzJUmAIbvvxAK5bgJv9ltOmEssIBRWD1cpqd8dD0Lo6CIUNEYCqtsfAHgm/U1KHQsVoHy2nFpFSDpBjjVSpBZkaPfgpLmaKJuiP3x+pFb2rvhDUe2sIQFPphISlu7T2Vr3K5EGofr4zvhrz9+gNlt8jys/z5Nqw8FxnRbe2Wyzan9NYP6IQI2E3rk=" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x<5", "result": "x<\\frac{5}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}<\\frac{5}{2}" }, { "type": "step", "primary": "Simplify", "result": "x<\\frac{5}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s733bmpCSrrIdw9VoQUW3QZnqmU5rQtJ8zdtqJ0kjhpCmAihI8YpOa7YjcZorvECwsGTs6/dv6vqGrZei/KKb0d7Kgkg/fXeg5nSca4R/eriYKA/ZpOZ/69yR+tuSn6BgZ0hYzF6vxjSZoMSInCY/PPQbO95/2yiZOrh8Vdx/OHPtolA7ljGwKjl4uJ3NTxfZc0+cMFhWhkyzb1aDn8WqP9U4LVfFHk5To268IqjZ58KLSYaQPf7g/ckfz4eqmlhX1OSsh7K0N7TukStXuv1XfrJWtRhIg+0KEyyHjCzeI3B/QaipHbr1nmdWpXYzbn83Bys4JCfXqVwFRIPJNMoUBh3rYYacq0E3WuEr1HGZvyrCREJa0Ou6rrvOkQ7oxS7UugIoSPGKTmu2I3GaK7xAsLDbcpGIV7EJ/S1nHIdsy+v0=" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$2x-5>0:{\\quad}x>\\frac{5}{2}$$", "input": "2x-5>0", "steps": [ { "type": "interim", "title": "Move $$5\\:$$to the right side", "input": "2x-5>0", "result": "2x>5", "steps": [ { "type": "step", "primary": "Add $$5$$ to both sides", "result": "2x-5+5>0+5" }, { "type": "step", "primary": "Simplify", "result": "2x>5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EnpPCbVNiLoYzx6Hs0PDcRyOUtErWNRXbTqqjj69E3v0/0aIMjdGoc6L+9BkyFeXRKvwpCsMm/dIDWKdnSUJtD8HtGQmCKxvJmlqmGE4mBmTDp38YGC88uSLUJ2/3TjNBII9y7a9iu65XQd3qNxlcHOuJioD0NRqN1Eah09L4f+/AqjyyjhUTD50HesEa0Dw7qz0Y/8NSmMUV1FULviXVKU5YxDLzW4dJ11H1VcrdkO0liQlMRTzJUmAIbvvxAK5bgJv9ltOmEssIBRWD1cpqd8dD0Lo6CIUNEYCqtsfAHgm/U1KHQsVoHy2nFpFSDpBjjVSpBZkaPfgpLmaKJuiP3x+pFb2rvhDUe2sIQFPphISlu7T2Vr3K5EGofr4zvhrz9+gNlt8jys/z5Nqw8FxnRbe2Wyzan9NYP6IQI2E3rk=" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x>5", "result": "x>\\frac{5}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}>\\frac{5}{2}" }, { "type": "step", "primary": "Simplify", "result": "x>\\frac{5}{2}" } 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"primary": "Subtract $$5$$ from both sides", "result": "3x+5-5=0-5" }, { "type": "step", "primary": "Simplify", "result": "3x=-5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74VFwlUYsxBf7of6zn1qlAxyOUtErWNRXbTqqjj69E3v0/0aIMjdGoc6L+9BkyFeXRKvwpCsMm/dIDWKdnSUJtBnIVaXPR88wDGXvKg8nszC8VmH5b+IMwMCjiIIju0VR3CmpoWEZ6Xx6oiIfKYR080iu613x5IyBSj4elHIXAAyl8j/x8cK2yFAJzw6WzJJDOFnYARhZt2YOCbDu1i5WSAwKAu7Z3tN/RgLPjlzyRw8i5OoNNprqQV2UgQUGESqFlvjhqQs5XJJkHv2GQUE6glcrAOsROfiVgP+wtWyh0afthvwKEoR3ksHNHbCkrb6kcwDIfxbr/Pq2vWjsXBdpmP+Gc7uNDn5rFMiDyv69niKS9nJYJSn323kBOgLLJJSHvRegFJlwIEwS0zZoQlFFa39YihiGR/khifj29p6+pso=" } }, { "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}" } ], 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"3x<-5" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ELDi/+du/aoxETcVlr8ikRyOUtErWNRXbTqqjj69E3v0/0aIMjdGoc6L+9BkyFeXRKvwpCsMm/dIDWKdnSUJtM3smYu2CGGw69JPvq6WDAq8VmH5b+IMwMCjiIIju0VR3CmpoWEZ6Xx6oiIfKYR080iu613x5IyBSj4elHIXAAyl8j/x8cK2yFAJzw6WzJJDOFnYARhZt2YOCbDu1i5WSAwKAu7Z3tN/RgLPjlzyRw8i5OoNNprqQV2UgQUGESqFlvjhqQs5XJJkHv2GQUE6glcrAOsROfiVgP+wtWyh0afthvwKEoR3ksHNHbCkrb6kcwDIfxbr/Pq2vWjsXBdpmP+Gc7uNDn5rFMiDyv69niKS9nJYJSn323kBOgLLJJSHvRegFJlwIEwS0zZoQlFFa39YihiGR/khifj29p6+pso=" } }, { "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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"x<8" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WfavNv+lTfnRTeELUPpddZN1pXT08zEQpn0WJ6CFMXCJupyMM6rmjOnkWROas+zAIv3pLYfKyQbKCshhkaZHUXf2GL05DDy6G+EIGVaKxxKJLbrsy9PUkt1PkbIMRxhVzPywKoVbgpEyLnOm1p35SJY8wUJivS+5lq5E9Saqzvxo4zDRBav7305um9/DE4BM3oBQia5lzwRgyTQp19wPYU8YrmSRejouV6cFhNwTUMDU3KAY4kDHilM6r45aMYmhRT6FvIpi+SIc8+o3gR8t1Ytduin5SBlz49E+ukld6ZgCJ1lQ1FAwVjR6mDWtuTH5DgDlk1i/4yAtEn/PcFV4wKN6Hv6MoTMtvtU0IQwXdn+7yfxP2MPOrJ8QyYBzj469+kz6Wg520T6srp4V+IOJLKV2iqUFTrKglIp5m/m6F0k=" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-8>0:{\\quad}x>8$$", "input": "x-8>0", "steps": [ { "type": "interim", "title": "Move $$8\\:$$to the right side", "input": "x-8>0", "result": "x>8", "steps": [ { "type": "step", "primary": "Add $$8$$ to both sides", "result": "x-8+8>0+8" }, { "type": "step", "primary": "Simplify", "result": "x>8" } ], 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both sides", "result": "x-2+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "x=2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vllAWdnRREEdiAcj4A0dPpN1pXT08zEQpn0WJ6CFMXCzpow5G3NsgikccJImBCG8Iv3pLYfKyQbKCshhkaZHUfi1QU7tgppKNJd7Fd5Q9rqJLbrsy9PUkt1PkbIMRxhVzPywKoVbgpEyLnOm1p35SJY8wUJivS+5lq5E9Saqzvxo4zDRBav7305um9/DE4BM3oBQia5lzwRgyTQp19wPYU8YrmSRejouV6cFhNwTUMDU3KAY4kDHilM6r45aMYmhRT6FvIpi+SIc8+o3gR8t1Ytduin5SBlz49E+ukld6ZgCJ1lQ1FAwVjR6mDWtuTH5DgDlk1i/4yAtEn/PcFV4wKN6Hv6MoTMtvtU0IQwXdn9M15GQap8jJgC1YN0KNA/e+kz6Wg520T6srp4V+IOJLKV2iqUFTrKglIp5m/m6F0k=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x-2<0:{\\quad}x<2$$", "input": "x-2<0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2<0", "result": "x<2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": 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"type": "step", "primary": "Simplify", "result": "x>2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PPSr9mCctJp+9U0OjE9Qp5N1pXT08zEQpn0WJ6CFMXCzpow5G3NsgikccJImBCG8Iv3pLYfKyQbKCshhkaZHURf4Pz4mpVMYKYf8U3xTtduJLbrsy9PUkt1PkbIMRxhVzPywKoVbgpEyLnOm1p35SJY8wUJivS+5lq5E9Saqzvxo4zDRBav7305um9/DE4BM3oBQia5lzwRgyTQp19wPYU8YrmSRejouV6cFhNwTUMDU3KAY4kDHilM6r45aMYmhRT6FvIpi+SIc8+o3gR8t1Ytduin5SBlz49E+ukld6ZgCJ1lQ1FAwVjR6mDWtuTH5DgDlk1i/4yAtEn/PcFV4wKN6Hv6MoTMtvtU0IQwXdn9M15GQap8jJgC1YN0KNA/e+kz6Wg520T6srp4V+IOJLKV2iqUFTrKglIp5m/m6F0k=" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "step", "primary": "Find singularity points" }, { "type": "interim", "title": "Find the zeros of the denominator $$\\left(3x+5\\right)\\left(x-8\\right)\\left(x-2\\right):{\\quad}x=-\\frac{5}{3},\\:x=8,\\:x=2$$", "input": 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"pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74VFwlUYsxBf7of6zn1qlAxyOUtErWNRXbTqqjj69E3v0/0aIMjdGoc6L+9BkyFeXRKvwpCsMm/dIDWKdnSUJtBnIVaXPR88wDGXvKg8nszC8VmH5b+IMwMCjiIIju0VR3CmpoWEZ6Xx6oiIfKYR080iu613x5IyBSj4elHIXAAyl8j/x8cK2yFAJzw6WzJJDOFnYARhZt2YOCbDu1i5WSAwKAu7Z3tN/RgLPjlzyRw8i5OoNNprqQV2UgQUGESqFlvjhqQs5XJJkHv2GQUE6glcrAOsROfiVgP+wtWyh0afthvwKEoR3ksHNHbCkrb6kcwDIfxbr/Pq2vWjsXBdpmP+Gc7uNDn5rFMiDyv69niKS9nJYJSn323kBOgLLJJSHvRegFJlwIEwS0zZoQlFFa39YihiGR/khifj29p6+pso=" } }, { "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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