2x+8>= 8(x-5)>7(x+3)-x

{ "query": { "display": "$$2x+8\\ge\\:8\\left(x-5\\right)>7\\left(x+3\\right)-x$$", "symbolab_question": "EQUATION#2x+8\\ge 8(x-5)>7(x+3)-x" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Inequalities", "subTopic": "CompoundInequalitySolver", "default": "\\mathrm{No\\:Solution}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2x+8\\ge\\:8\\left(x-5\\right)>7\\left(x+3\\right)-x{\\quad:\\quad}$$No Solution", "input": "2x+8\\ge\\:8\\left(x-5\\right)>7\\left(x+3\\right)-x", "steps": [ { "type": "step", "primary": "If $$a\\ge\\:u>b\\:$$then $$a\\ge\\:u\\land\\:u>b$$", "result": "2x+8\\ge\\:8\\left(x-5\\right)\\land\\:8\\left(x-5\\right)>7\\left(x+3\\right)-x" }, { "type": "interim", "title": "$$2x+8\\ge\\:8\\left(x-5\\right){\\quad:\\quad}x\\le\\:8$$", "input": "2x+8\\ge\\:8\\left(x-5\\right)", "steps": [ { "type": "interim", "title": "Expand $$8\\left(x-5\\right):{\\quad}8x-40$$", "input": "8\\left(x-5\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=8,\\:b=x,\\:c=5$$" ], "result": "=8x-8\\cdot\\:5", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:5=40$$", "result": "=8x-40" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WUKTXSXih/VrMsNBhFFu9wsBmUOGgB0VAXoAXik9qC9EoApuujwRq/Ms3wa4gNT89ZVSO0B13oSpK+zJfqv7kB7MWKURN+43KCOzRc+RXaEKbXYLThMma1MQpXiorIlI" } }, { "type": "step", "result": "2x+8\\ge\\:8x-40" }, { "type": "interim", "title": "Move $$8\\:$$to the right side", "input": "2x+8\\ge\\:8x-40", "result": "2x\\ge\\:8x-48", "steps": [ { "type": "step", "primary": "Subtract $$8$$ from both sides", "result": "2x+8-8\\ge\\:8x-40-8" }, { "type": "step", "primary": "Simplify", "result": "2x\\ge\\:8x-48" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$8x\\:$$to the left side", "input": "2x\\ge\\:8x-48", "result": "-6x\\ge\\:-48", "steps": [ { "type": "step", "primary": "Subtract $$8x$$ from both sides", "result": "2x-8x\\ge\\:8x-48-8x" }, { "type": "step", "primary": "Simplify", "result": "-6x\\ge\\:-48" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Multiply both sides by $$-1$$", "input": "-6x\\ge\\:-48", "result": "6x\\le\\:48", "steps": [ { "type": "step", "primary": "Multiply both sides by -1 (reverse the inequality)", "result": "\\left(-6x\\right)\\left(-1\\right)\\le\\:\\left(-48\\right)\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "6x\\le\\:48" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70jjeS+7S91w1EYZ6Wt/6Go3vu8LX5kAGCoNRmoPDo3SxzEQAkj1QaYXDu9z2KrgR4WLGPOzAiVcWcvN5piR4L2RWA4CcN7TiqPLbvvTZ52xKCb0Aw62frMCpj1vOtu3GT6E+Z+lYFfRqAnQ26sDvnCwtYv+gYP75aLTZcT7K4J2Rfpj0VM9FjCQFt3Ycp+umj8L89Wlf8jbe4o1i7wFeoIWUKKt9rgNT6q1Dvse3Sq7kXkqvVQ0zyEKMC6tVox59h2lnLhfgYibgjonQzk7m+G6uRWZBrrLPfEFIC3Te7pM/r3xh7fKzcD/kct06ILUXaS9NV4InuUmO1C9/jyi/V5tnuYPhPiE5kjSssP4vT+cEfWloc8TWIawViUVLHvB7p+llO8m1JUsHsxYSyr5CjVb7qlKGJCpgSun0kdnDZYJH6n09x0sZXZjbeq1HH78kgsqwDvczEA2coxIljO6iORzS2phEsol5BFTVOAXt8l5V96z+yd/NTgi3X+wu1lkQ72wZm7kDUxdE6YSmfEbr2sGQWPvmw/6Rl39t7klTO98Uy14I6ZTWoUzXAJYyzItHvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6x\\le\\:48", "result": "x\\le\\:8", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6x}{6}\\le\\:\\frac{48}{6}" }, { "type": "step", "primary": "Simplify", "result": "x\\le\\:8" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$8\\left(x-5\\right)>7\\left(x+3\\right)-x{\\quad:\\quad}x>\\frac{61}{2}$$", "input": "8\\left(x-5\\right)>7\\left(x+3\\right)-x", "steps": [ { "type": "interim", "title": "Expand $$8\\left(x-5\\right):{\\quad}8x-40$$", "input": "8\\left(x-5\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=8,\\:b=x,\\:c=5$$" ], "result": "=8x-8\\cdot\\:5", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:5=40$$", "result": "=8x-40" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WUKTXSXih/VrMsNBhFFu9wsBmUOGgB0VAXoAXik9qC9EoApuujwRq/Ms3wa4gNT89ZVSO0B13oSpK+zJfqv7kB7MWKURN+43KCOzRc+RXaEKbXYLThMma1MQpXiorIlI" } }, { "type": "interim", "title": "Expand $$7\\left(x+3\\right)-x:{\\quad}6x+21$$", "input": "7\\left(x+3\\right)-x", "steps": [ { "type": "interim", "title": "Expand $$7\\left(x+3\\right):{\\quad}7x+21$$", "input": "7\\left(x+3\\right)", "result": "=7x+21-x", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=7,\\:b=x,\\:c=3$$" ], "result": "=7x+7\\cdot\\:3", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$7\\cdot\\:3=21$$", "result": "=7x+21" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7U2Hvv/IaN4BJRkhRKwLoTXWD310L1+P2yDQQfMEhENFtFTsvMYeaPaoNPUgk+8mo8LfSxJ+0AgVLpCSnLX0iStRJlYQrdBn02pAQzmwbYVTiAEmXhYw7WsDRrfT9tRiW" } }, { "type": "interim", "title": "Simplify $$7x+21-x:{\\quad}6x+21$$", "input": "7x+21-x", "result": "=6x+21", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=7x-x+21" }, { "type": "step", "primary": "Add similar elements: $$7x-x=6x$$", "result": "=6x+21" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kzJDzc/X1kiV2aarWrfIa08iHIAOx4OsbU025/ywWBgF0f9e9MOLJfBD6LX95TzkeFr9fJWrn4Z0QR9jTkKwM0DLjX7DcYtB54q3geRNejHv7aEgwhgwQKhnpyvqEpv8vzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "result": "8x-40>6x+21" }, { "type": "interim", "title": "Move $$40\\:$$to the right side", "input": "8x-40>6x+21", "result": "8x>6x+61", "steps": [ { "type": "step", "primary": "Add $$40$$ to both sides", "result": "8x-40+40>6x+21+40" }, { "type": "step", "primary": "Simplify", "result": "8x>6x+61" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$6x\\:$$to the left side", "input": "8x>6x+61", "result": "2x>61", "steps": [ { "type": "step", "primary": "Subtract $$6x$$ from both sides", "result": "8x-6x>6x+61-6x" }, { "type": "step", "primary": "Simplify", "result": "2x>61" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ac7C94Jyg6I6BsqoRavdxIGk3UVA43yzR6zHaAMYbX7+UpjRogLJqqPzFgp5n/s288UxTzlGSTEJ2AYfegcR+/2SfWzaRsW/bMiIofqcZulbwVz5T7/Dyunz8G+H2e0kUsD+pIkOCIZyIU4q8atffPpdu80I8/0M2UIpmdiK+fxHpZfRm/EBDqU1remlDehgt8ZQzL5RHmLvzqO1uUVUUTR59k7NckglC0H4kGCfrGiH7U/MsnOpPfk01dDyeXAssmDv4bNUTCmqLSGwavYzuvvhfZZlsZmANAFCeZqDcFhk/2DE6344aSXGc0jSlPxfUVzPX+B3S8n2NYuU1+1vaNMqohqs5dK58z4VC4uF1sPkU5TkMMWES51yFsTkJT6lflB44+cAhbl05D7N1o4XnYUDxBuztWdWdEmOXBe3LLpWPNqzEhkP5qD6ioNU4s9fi9c/nHGhIqvPRpttdSi2lpJj/5xKKdB5FHwO4G/fHn8mxF5pxAejgMBdXf8NHl+mk3dK9hN6pPkzVNfz8kjct+cAmWY31CEn6duy4rD8zNuTu7I3xiOi2nWH/AU9cMW4sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x>61", "result": "x>\\frac{61}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}>\\frac{61}{2}" }, { "type": "step", "primary": "Simplify", "result": "x>\\frac{61}{2}" } ], "meta": { 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"result": "\\mathrm{No\\:Solution}", "steps": [ { "type": "step", "primary": "The intersection of two intervals is the set of numbers which are in both intervals<br/>$$x\\le\\:8\\quad$$and$$\\quad\\:x>\\frac{61}{2}$$", "image": "/images/interval?expression=%28y_%7B0%7D%5Cle+8%29%5Cland+%28y_%7B0%7D%3E%5Cfrac%7B61%7D%7B2%7D%29", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "Merge Overlapping Intervals 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQYGckMT/cy9E3O0ylgEE54l1tGTu2AjPk9jLJa0kTB9sVFP5aPp8ShpnPl75aN0HimMskdg7nehpYklu3te90+/yeSBbiqMrAOOLmT6vrKbPyKzkhXryWBV4Bv2UwbPXs" } } ], "meta": { "solvingClass": "Inequalities" } }, "meta": { "showVerify": true } }