5/(x-6)> 3/(x+2)

{ "query": { "display": "$$\\frac{5}{x-6}>\\frac{3}{x+2}$$", "symbolab_question": "EQUATION#\\frac{5}{x-6}>\\frac{3}{x+2}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Inequalities", "subTopic": "RationalIneqSolver", "default": "-14<x<-2\\lor x>6", "interval": "(-14,-2)\\cup (6,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{5}{x-6}>\\frac{3}{x+2}{\\quad:\\quad}-14<x<-2\\lor\\:x>6$$", "input": "\\frac{5}{x-6}>\\frac{3}{x+2}", "steps": [ { "type": "interim", "title": "Rewrite in standard form", "input": "\\frac{5}{x-6}>\\frac{3}{x+2}", "result": "\\frac{2x+28}{\\left(x-6\\right)\\left(x+2\\right)}>0", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{3}{x+2}$$ from both sides", "result": "\\frac{5}{x-6}-\\frac{3}{x+2}>\\frac{3}{x+2}-\\frac{3}{x+2}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{5}{x-6}-\\frac{3}{x+2}>0" }, { "type": "interim", "title": "Simplify $$\\frac{5}{x-6}-\\frac{3}{x+2}:{\\quad}\\frac{2x+28}{\\left(x-6\\right)\\left(x+2\\right)}$$", "input": "\\frac{5}{x-6}-\\frac{3}{x+2}", "result": "\\frac{2x+28}{\\left(x-6\\right)\\left(x+2\\right)}>0", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$x-6,\\:x+2:{\\quad}\\left(x-6\\right)\\left(x+2\\right)$$", "input": "x-6,\\:x+2", "steps": [ { "type": "definition", "title": "Lowest Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b\\:$$is the smallest multiplier that is divisible by both $$a$$ and $$b$$" }, { "type": "step", "primary": "Compute an expression comprised of factors that appear either in $$x-6$$ or $$x+2$$", "result": "=\\left(x-6\\right)\\left(x+2\\right)" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$\\left(x-6\\right)\\left(x+2\\right)$$" }, { "type": "step", "primary": "For $$\\frac{5}{x-6}:\\:$$multiply the denominator and numerator by $$x+2$$", "result": "\\frac{5}{x-6}=\\frac{5\\left(x+2\\right)}{\\left(x-6\\right)\\left(x+2\\right)}" }, { "type": "step", "primary": "For $$\\frac{3}{x+2}:\\:$$multiply the denominator and numerator by $$x-6$$", "result": "\\frac{3}{x+2}=\\frac{3\\left(x-6\\right)}{\\left(x+2\\right)\\left(x-6\\right)}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{5\\left(x+2\\right)}{\\left(x-6\\right)\\left(x+2\\right)}-\\frac{3\\left(x-6\\right)}{\\left(x+2\\right)\\left(x-6\\right)}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{5\\left(x+2\\right)-3\\left(x-6\\right)}{\\left(x-6\\right)\\left(x+2\\right)}" }, { "type": "interim", "title": "Expand $$5\\left(x+2\\right)-3\\left(x-6\\right):{\\quad}2x+28$$", "input": "5\\left(x+2\\right)-3\\left(x-6\\right)", "result": "=\\frac{2x+28}{\\left(x-6\\right)\\left(x+2\\right)}", "steps": [ { "type": "interim", "title": "Expand $$5\\left(x+2\\right):{\\quad}5x+10$$", "input": "5\\left(x+2\\right)", "result": "=5x+10-3\\left(x-6\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=5,\\:b=x,\\:c=2$$" ], "result": "=5x+5\\cdot\\:2", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:2=10$$", "result": "=5x+10" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76A+KsZpx+yOkL7MR3ijPEXWD310L1+P2yDQQfMEhENFh0JkxGu7a3f99+stIcEFE8LfSxJ+0AgVLpCSnLX0iSqiuiDY5YPj0g2UlxdiIOODiAEmXhYw7WsDRrfT9tRiW" } }, { "type": "interim", "title": "Expand $$-3\\left(x-6\\right):{\\quad}-3x+18$$", "input": "-3\\left(x-6\\right)", "result": "=5x+10-3x+18", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-3,\\:b=x,\\:c=6$$" ], "result": "=-3x-\\left(-3\\right)\\cdot\\:6", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-3x+3\\cdot\\:6" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:6=18$$", "result": "=-3x+18" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78am5eTMGAjCrURiX3unVrAOfOVs9mPIqDLV5QIWwt3n6ZVaQT0jpALkWAgiP/oQF72wZm7kDUxdE6YSmfEbr2mPWFmYbm6gt7L0gsaIfTfWASliZ8tPk1XaFnknKtvWB" } }, { "type": "interim", "title": "Simplify $$5x+10-3x+18:{\\quad}2x+28$$", "input": "5x+10-3x+18", "result": "=2x+28", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=5x-3x+10+18" }, { "type": "step", "primary": "Add similar elements: $$5x-3x=2x$$", "result": "=2x+10+18" }, { "type": "step", "primary": "Add the numbers: $$10+18=28$$", "result": "=2x+28" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/6xBjnRBTcY544RsPFBrMS061ljBSPJeENOw2efoSWtFklkQ4ppWNoN7hrFol5AVrhHjQYEmty6yL8d9wP48FPxtgDRvpXZFJM5ns6z004PgPdnSw5AP7hZuF7jUUHjwvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Geometry Write In Standard Form Title 0Eq" } }, { "type": "interim", "title": "Factor $$\\frac{2x+28}{\\left(x-6\\right)\\left(x+2\\right)}:{\\quad}\\frac{2\\left(x+14\\right)}{\\left(x-6\\right)\\left(x+2\\right)}$$", "input": "\\frac{2x+28}{\\left(x-6\\right)\\left(x+2\\right)}", "result": "\\frac{2\\left(x+14\\right)}{\\left(x-6\\right)\\left(x+2\\right)}>0", "steps": [ { "type": "interim", "title": "Factor $$2x+28:{\\quad}2\\left(x+14\\right)$$", "input": "2x+28", "result": "=\\frac{2\\left(x+14\\right)}{\\left(x-6\\right)\\left(x+2\\right)}", "steps": [ { "type": "interim", "title": "Factor out common term $$2:{\\quad}2\\left(x+14\\right)$$", "input": "2x+28", "steps": [ { "type": "step", "primary": "Rewrite $$28$$ as $$2\\cdot\\:14$$", "result": "=2x+2\\cdot\\:14" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(x+14\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=2\\left(x+14\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{\\frac{2\\left(x+14\\right)}{\\left(x-6\\right)\\left(x+2\\right)}}{2}>\\frac{0}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{x+14}{\\left(x-6\\right)\\left(x+2\\right)}>0" }, { "type": "interim", "title": "Identify the intervals", "result": "-14<x<-2\\lor\\:x>6", "steps": [ { "type": "step", "primary": "Find the signs of the factors of $$\\frac{x+14}{\\left(x-6\\right)\\left(x+2\\right)}$$" }, { "type": "interim", "title": "Find the signs of $$x+14$$", "steps": [ { "type": "interim", "title": "$$x+14=0:{\\quad}x=-14$$", "input": "x+14=0", "steps": [ { "type": "interim", "title": "Move $$14\\:$$to the right side", "input": "x+14=0", "result": "x=-14", "steps": [ { "type": "step", "primary": "Subtract $$14$$ from both sides", "result": "x+14-14=0-14" }, { "type": "step", "primary": "Simplify", "result": "x=-14" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x+14<0:{\\quad}x<-14$$", "input": "x+14<0", "steps": [ { "type": "interim", "title": "Move $$14\\:$$to the right side", "input": "x+14<0", "result": "x<-14", "steps": [ { "type": "step", "primary": "Subtract $$14$$ from both sides", "result": "x+14-14<0-14" }, { "type": "step", "primary": "Simplify", "result": "x<-14" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x+14>0:{\\quad}x>-14$$", "input": "x+14>0", "steps": [ { "type": "interim", "title": "Move $$14\\:$$to the right side", "input": "x+14>0", "result": "x>-14", "steps": [ { "type": "step", "primary": "Subtract $$14$$ from both sides", "result": "x+14-14>0-14" }, { "type": "step", "primary": "Simplify", "result": "x>-14" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x-6$$", "steps": [ { "type": "interim", "title": "$$x-6=0:{\\quad}x=6$$", "input": "x-6=0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "x-6=0", "result": "x=6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "x-6+6=0+6" }, { "type": "step", "primary": "Simplify", "result": "x=6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$x-6<0:{\\quad}x<6$$", "input": "x-6<0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "x-6<0", "result": "x<6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "x-6+6<0+6" }, { "type": "step", "primary": "Simplify", "result": "x<6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "type": "interim", "title": "$$x-6>0:{\\quad}x>6$$", "input": "x-6>0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "x-6>0", "result": "x>6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "x-6+6>0+6" }, { "type": "step", "primary": "Simplify", "result": "x>6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } } ], "meta": { "interimType": "Find Sign 1Eq" } }, { "type": "interim", "title": "Find the signs of $$x+2$$", "steps": [ { "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": "Subtract $$2$$ from both sides", "result": "x+2-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "x=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/Ry78kxdougw927cPeeMgpN1pXT08zEQpn0WJ6CFMXCzpow5G3NsgikccJImBCG8Iv3pLYfKyQbKCshhkaZHURJSyOxsZoLtDFlQE5qDAKSyiHU3VsxZ0Jvlgz1Dh/2W0zIpeqOKDWh7ydUbxgKhdXr3g+Hzr2XIOhBR2yznfms+YM902o61cMfuzaDvZcmEJfZxvFZvfiMgWozzdsoT74sIFEGGlIvsQPM44IkyyuRBVBNkVSABMbP/i1bzzdBt2Oyx6db4s6omnIZraJUd12T/YMTrfjhpJcZzSNKU/F+DZp63hBGhjskvuDfrfb9Z+M47wUwofebcc2b4hlDDCXgdyTSIALeKJHOd0bpb+aksXf9zkGMZrK+UegK8hFqWzbVvsbe2FwB551F3zoabdhcpTddhXB6PpdnfagbEv9x+c/SZGKJlPT9WBkxURag3TeQKHeh69S6dnv9vSoUoFG5JhQGIxQh0RzVrWHgpcfP7pDrkjdeHr9R1cIBxlUa66N/nbfo9X4bAb2s+hgXibQ==" } } ], "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": "Subtract $$2$$ from both sides", "result": "x+2-2<0-2" }, { "type": "step", "primary": "Simplify", "result": "x<-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Inequalities", "interimType": "Inequalities" } }, { "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": "Subtract $$2$$ from both sides", "result": "x+2-2>0-2" }, { "type": "step", "primary": "Simplify", "result": "x>-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "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 $$\\left(x-6\\right)\\left(x+2\\right):{\\quad}x=6,\\:x=-2$$", "input": "\\left(x-6\\right)\\left(x+2\\right)=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "x-6=0\\lor\\:x+2=0" }, { "type": "interim", "title": "Solve $$x-6=0:{\\quad}x=6$$", "input": "x-6=0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "x-6=0", "result": "x=6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "x-6+6=0+6" }, { "type": "step", "primary": "Simplify", "result": "x=6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$x+2=0:{\\quad}x=-2$$", "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": "Subtract $$2$$ from both sides", "result": "x+2-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "x=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=6,\\:x=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Find Denom Zeroes Title 1Eq" } }, { "type": "step", "primary": "Summarize in a table:", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|c|c|}\\hline &x<-14&x=-14&-14<x<-2&x=-2&-2<x<6&x=6&x>6\\\\\\hline x+14&-&0&+&+&+&+&+\\\\\\hline x-6&-&-&-&-&-&0&+\\\\\\hline x+2&-&-&-&0&+&+&+\\\\\\hline \\frac{x+14}{(x-6)(x+2)}&-&0&+&\\mathrm{Undefined}&-&\\mathrm{Undefined}&+\\\\\\hline \\end{array}$$" ] }, { "type": "step", "primary": "Identify the intervals that satisfy the required condition: $$>\\:0$$", "result": "-14<x<-2\\lor\\:x>6" } ], "meta": { "interimType": "Identify The Intervals NoCol 0Eq" } } ], "meta": { "solvingClass": "Inequalities", "practiceLink": "/practice/inequalities-practice#area=main&subtopic=Rational", "practiceTopic": "Rational Inequalities" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "x=-14", "displayFormula": "x=-14", "attributes": { "color": "GRAY", "lineType": "DASH", "isAsymptote": false } }, { "evalFormula": "x=-2", "displayFormula": "x=-2", "attributes": { "color": "GRAY", "lineType": "DASH", "isAsymptote": false } }, { "evalFormula": "x=6", "displayFormula": "x=6", "attributes": { "color": "GRAY", "lineType": "DASH", "isAsymptote": false } } ] }, "fills": [ { "ranges": [ { "fst": "-14.0", "snd": "-2.0" }, { "fst": "6.0", "snd": "Infinity" } ], "funcIndices": [], "funcs": [], "xIneq": true, "yIneq": false, "twoVar": false, "trueAboveLine": false, "color": "rgba(171, 181, 235, 0.3)" } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "-14<x<-2\\lor x>6", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -18, "xMax": 10, "yMin": -2, "yMax": 8 }, "intervals": [ { "mind": -14, "maxd": -2, "minInclusive": false, "maxInclusive": false, "maxFinite": true, "minFinite": true, "isAllR": false, "isPoint": false }, { "mind": 6, "minInclusive": false, "maxInclusive": false, "maxFinite": false, "minFinite": true, "isAllR": false, "isPoint": false } ], "imageAPI": "-14<x<-2\\lor x>6" }, "showViewLarger": true } }, "meta": { "showVerify": true } }