{
"query": {
"display": "$$\\frac{x+6}{x^{2}+1}<0$$",
"symbolab_question": "EQUATION#\\frac{x+6}{x^{2}+1}<0"
},
"solution": {
"level": "PERFORMED",
"subject": "Algebra",
"topic": "Inequalities",
"subTopic": "RationalIneqSolver",
"default": "x<-6",
"interval": "(-\\infty ,-6)",
"meta": {
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}
},
"steps": {
"type": "interim",
"title": "$$\\frac{x+6}{x^{2}+1}<0{\\quad:\\quad}x<-6$$",
"input": "\\frac{x+6}{x^{2}+1}<0",
"steps": [
{
"type": "interim",
"title": "Identificar los intervalos",
"result": "x<-6",
"steps": [
{
"type": "step",
"primary": "Encontrar los signos de los factores de $$\\frac{x+6}{x^{2}+1}$$"
},
{
"type": "interim",
"title": "Encontrar los signos de $$x+6$$",
"steps": [
{
"type": "interim",
"title": "$$x+6=0:{\\quad}x=-6$$",
"input": "x+6=0",
"steps": [
{
"type": "interim",
"title": "Desplace $$6\\:$$a la derecha",
"input": "x+6=0",
"result": "x=-6",
"steps": [
{
"type": "step",
"primary": "Restar $$6$$ de ambos lados",
"result": "x+6-6=0-6"
},
{
"type": "step",
"primary": "Simplificar",
"result": "x=-6"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
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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": "Desplace $$6\\:$$a la derecha",
"input": "x+6<0",
"result": "x<-6",
"steps": [
{
"type": "step",
"primary": "Restar $$6$$ de ambos lados",
"result": "x+6-6<0-6"
},
{
"type": "step",
"primary": "Simplificar",
"result": "x<-6"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eosBx8i32BG8zPLG6snt1ZN1pXT08zEQpn0WJ6CFMXBU+fhk3HPlBZCpAQKU+c07PlRMTLc8dpF+T7mmaRQSVq0IvTf8nuxwKi5hRjeKrt6JLbrsy9PUkt1PkbIMRxhVzPywKoVbgpEyLnOm1p35SJY8wUJivS+5lq5E9Saqzvw+EYDnTlI3BaBy6CC+VjzLsBIDdmsik4IgLuk24+fXheCDCH2DcDNialMuvy7Bnwnk+UY9JBE5v/YSZ6s+GczznYqnACvNSqOPyqchT4gmfHabZwi5JytN/1ufmQiSimkGMgSS9o9O+UehR1a3T1ITWnYMM7mOiW9HKIEiHilarlHG4pyteBqJ+X90AXJ/ySo1b0arRRaesUXU9U12Bl/4usYwrU5d1Hf8HHZ1+D9TYJe1d9OHFIbfl94J3fiBKM8="
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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": "Desplace $$6\\:$$a la derecha",
"input": "x+6>0",
"result": "x>-6",
"steps": [
{
"type": "step",
"primary": "Restar $$6$$ de ambos lados",
"result": "x+6-6>0-6"
},
{
"type": "step",
"primary": "Simplificar",
"result": "x>-6"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kuK2L2Uce+PeK0DN01f7R5N1pXT08zEQpn0WJ6CFMXBU+fhk3HPlBZCpAQKU+c07PlRMTLc8dpF+T7mmaRQSVj1+FSxKYpGZEXooSD21hFyJLbrsy9PUkt1PkbIMRxhVzPywKoVbgpEyLnOm1p35SJY8wUJivS+5lq5E9Saqzvw+EYDnTlI3BaBy6CC+VjzLsBIDdmsik4IgLuk24+fXheCDCH2DcDNialMuvy7Bnwnk+UY9JBE5v/YSZ6s+GczznYqnACvNSqOPyqchT4gmfHabZwi5JytN/1ufmQiSimkGMgSS9o9O+UehR1a3T1ITWnYMM7mOiW9HKIEiHilarlHG4pyteBqJ+X90AXJ/ySo1b0arRRaesUXU9U12Bl/4usYwrU5d1Hf8HHZ1+D9TYJe1d9OHFIbfl94J3fiBKM8="
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],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Inequalities"
}
}
],
"meta": {
"interimType": "Find Sign 1Eq"
}
},
{
"type": "interim",
"title": "Encontrar los signos de $$x^{2}+1:{\\quad}$$Positivo para todos los valores reales",
"input": "x^{2}+1>0",
"steps": [
{
"type": "interim",
"title": "Desplace $$1\\:$$a la derecha",
"input": "x^{2}+1>0",
"result": "x^{2}>-1",
"steps": [
{
"type": "step",
"primary": "Restar $$1$$ de ambos lados",
"result": "x^{2}+1-1>0-1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "x^{2}>-1"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
},
{
"type": "step",
"primary": "Si n es par, $$u^n\\:\\geq\\:0$$ para todo $$u$$",
"result": "\\mathrm{Verdadero\\:para\\:todo\\:}x"
}
],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Find Sign 1Eq"
}
},
{
"type": "step",
"primary": "Encontrar puntos de singularidad"
},
{
"type": "interim",
"title": "Encontrar los ceros del denominador $$x^{2}+1:{\\quad}$$Sin solución",
"input": "x^{2}+1=0",
"steps": [
{
"type": "interim",
"title": "Desplace $$1\\:$$a la derecha",
"input": "x^{2}+1=0",
"result": "x^{2}=-1",
"steps": [
{
"type": "step",
"primary": "Restar $$1$$ de ambos lados",
"result": "x^{2}+1-1=0-1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "x^{2}=-1"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
},
{
"type": "step",
"primary": "$$x^{2}$$ no puede ser negativo para $$x\\in\\mathbb{R}$$",
"result": "\\mathrm{Sin\\:solución\\:para}\\:x\\in\\mathbb{R}"
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Find Denom Zeroes Title 1Eq"
}
},
{
"type": "step",
"primary": "Resumir en una tabla:",
"secondary": [
"$$\\begin{array}{|c|c|c|c|}\\hline &x<-6&x=-6&x>-6\\\\\\hline x+6&-&0&+\\\\\\hline x^{2}+1&+&+&+\\\\\\hline \\frac{x+6}{x^{2}+1}&-&0&+\\\\\\hline \\end{array}$$"
]
},
{
"type": "step",
"primary": "Identificar los intervalos que cumplen la condición: $$<\\:0$$",
"result": "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=-6",
"displayFormula": "x=-6",
"attributes": {
"color": "GRAY",
"lineType": "DASH",
"isAsymptote": false
}
}
]
},
"fills": [
{
"ranges": [
{
"fst": "-Infinity",
"snd": "-6.0"
}
],
"funcIndices": [],
"funcs": [],
"xIneq": true,
"yIneq": false,
"twoVar": false,
"trueAboveLine": false,
"color": "rgba(171, 181, 235, 0.3)"
}
],
"functionChanges": [
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"finalFormulaLatex": [],
"plotTitle": "x<-6",
"paramsLatex": [],
"paramsReplacementsLatex": []
}
],
"localBoundingBox": {
"xMin": -8,
"xMax": 1.1428571428571428,
"yMin": -2,
"yMax": 8
},
"intervals": [
{
"maxd": -6,
"minInclusive": false,
"maxInclusive": false,
"maxFinite": true,
"minFinite": false,
"isAllR": false,
"isPoint": false
}
],
"imageAPI": "x<-6"
},
"showViewLarger": true
}
},
"meta": {
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}
}
Solución
Solución
+1
Notación de intervalos
Pasos de solución
Identificar los intervalos