-1/2 (4x+20)<-7(x+15/7)

{ "query": { "display": "$$-\\frac{1}{2}\\left(4x+20\\right)<-7\\left(x+\\frac{15}{7}\\right)$$", "symbolab_question": "EQUATION#-\\frac{1}{2}(4x+20)<-7(x+\\frac{15}{7})" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Inequalities", "subTopic": "Linear", "default": "x<-1", "interval": "(-\\infty ,-1)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$-\\frac{1}{2}\\left(4x+20\\right)<-7\\left(x+\\frac{15}{7}\\right){\\quad:\\quad}x<-1$$", "input": "-\\frac{1}{2}\\left(4x+20\\right)<-7\\left(x+\\frac{15}{7}\\right)", "steps": [ { "type": "interim", "title": "Expand $$-\\frac{1}{2}\\left(4x+20\\right):{\\quad}-2x-10$$", "input": "-\\frac{1}{2}\\left(4x+20\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:\\left(4x+20\\right)}{2}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(4x+20\\right)=4x+20$$", "input": "1\\cdot\\:\\left(4x+20\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(4x+20\\right)=\\left(4x+20\\right)$$", "result": "=\\left(4x+20\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=4x+20" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7n+M1PBFrfF7w/Q0Pb8V2NNMvHyY50dhXPFfrjcmooUhrWczDPFmHvnygQCovsCz0Dzeo0X+YTF2YzjGhjw5WZVhC1WJiMQoAj8s/UMaHtAI16OTYOz+c0TR6cTFJDN+J" } }, { "type": "step", "result": "=-\\frac{4x+20}{2}" }, { "type": "interim", "title": "Cancel $$\\frac{4x+20}{2}:{\\quad}2\\left(x+5\\right)$$", "input": "\\frac{4x+20}{2}", "result": "=-2\\left(x+5\\right)", "steps": [ { "type": "interim", "title": "Factor $$4x+20:{\\quad}4\\left(x+5\\right)$$", "input": "4x+20", "result": "=\\frac{4\\left(x+5\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=4x+4\\cdot\\:5" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=4\\left(x+5\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2\\left(x+5\\right)" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnuF4Dom3gwDZiAAFQ17tLTNGoPE9TME3q+OPmgkv2RQBuV8q5RiqmfJ5YYogdeOF4EFMST8lDZxn1Yq5HMKVTufNxfP+VYga2fBBEWdOHE5W73Q3y1IDpflMvDNGIUqOA==" } }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-2,\\:b=x,\\:c=5$$" ], "result": "=-2x+\\left(-2\\right)\\cdot\\:5", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-2x-2\\cdot\\:5" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=-2x-10" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74TqdRGDyv3oqElK9EevGS+n6dTvu0FJHVIvA/Fjxb/Haec1RZIwlCMvbTN26PYgr6ZvcSHthJ+x3ZgtR9gxl5lh0eio4FTI3JXph8YFx3d2CcvL2jSzw1VlRcGX4ka6q1FRMTm4ZnrI+nx2ZwjXMk+d89Zg0wPLXgNVaLS78d+4=" } }, { "type": "interim", "title": "Expand $$-7\\left(x+\\frac{15}{7}\\right):{\\quad}-7x-15$$", "input": "-7\\left(x+\\frac{15}{7}\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-7,\\:b=x,\\:c=\\frac{15}{7}$$" ], "result": "=-7x+\\left(-7\\right)\\frac{15}{7}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-7x-7\\cdot\\:\\frac{15}{7}" }, { "type": "interim", "title": "$$7\\cdot\\:\\frac{15}{7}=15$$", "input": "7\\cdot\\:\\frac{15}{7}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{15\\cdot\\:7}{7}" }, { "type": "step", "primary": "Cancel the common factor: $$7$$", "result": "=15" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7uxjTDNyZVStXsVC0PezoqSQ1HOvuArE/B2Hp4PADpW0JQJZuTAY5js+oqjdT8kslwqNUKf1tlpx+XShVSHQ2+q7E2pjI0TdNIzvBNHy3ya9W67WqHLm4VsnhCchZjfeFJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=-7x-15" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": 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