{
"query": {
"display": "$$\\left|x-5\\right|\\ge\\:3$$",
"symbolab_question": "EQUATION#\\left|x-5\\right|\\ge 3"
},
"solution": {
"level": "PERFORMED",
"subject": "Algebra",
"topic": "Inequalities",
"subTopic": "Absolute",
"default": "x\\le 2\\lor x\\ge 8",
"interval": "(-\\infty ,2]\\cup [8,\\infty )",
"meta": {
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},
"steps": {
"type": "interim",
"title": "$$\\left|x-5\\right|\\ge\\:3{\\quad:\\quad}x\\le\\:2\\lor\\:x\\ge\\:8$$",
"input": "\\left|x-5\\right|\\ge\\:3",
"steps": [
{
"type": "step",
"primary": "Apply absolute rule: If $$|u|\\:\\geq\\:a,\\:a\\:>\\:0\\:$$then $$u\\:\\leq\\:-a\\:\\quad\\:$$or$$\\quad\\:u\\:\\geq\\:a$$",
"result": "x-5\\le\\:-3\\lor\\:x-5\\ge\\:3"
},
{
"type": "interim",
"title": "$$x-5\\le\\:-3\\lor\\:x-5\\ge\\:3$$",
"input": "x-5\\le\\:-3\\lor\\:x-5\\ge\\:3",
"result": "x\\le\\:2\\lor\\:x\\ge\\:8",
"steps": [
{
"type": "interim",
"title": "$$x-5\\le\\:-3{\\quad:\\quad}x\\le\\:2$$",
"input": "x-5\\le\\:-3",
"steps": [
{
"type": "interim",
"title": "Move $$5\\:$$to the right side",
"input": "x-5\\le\\:-3",
"result": "x\\le\\:2",
"steps": [
{
"type": "step",
"primary": "Add $$5$$ to both sides",
"result": "x-5+5\\le\\:-3+5"
},
{
"type": "step",
"primary": "Simplify",
"result": "x\\le\\:2"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Inequalities"
}
},
{
"type": "interim",
"title": "$$x-5\\ge\\:3{\\quad:\\quad}x\\ge\\:8$$",
"input": "x-5\\ge\\:3",
"steps": [
{
"type": "interim",
"title": "Move $$5\\:$$to the right side",
"input": "x-5\\ge\\:3",
"result": "x\\ge\\:8",
"steps": [
{
"type": "step",
"primary": "Add $$5$$ to both sides",
"result": "x-5+5\\ge\\:3+5"
},
{
"type": "step",
"primary": "Simplify",
"result": "x\\ge\\:8"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Inequalities"
}
},
{
"type": "step",
"primary": "Combine the intervals",
"result": "x\\le\\:2\\lor\\:x\\ge\\:8"
}
],
"meta": {
"interimType": "N/A"
}
}
],
"meta": {
"solvingClass": "Inequalities"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"funcsToDraw": {
"funcs": [
{
"evalFormula": "x=2",
"displayFormula": "x=2",
"attributes": {
"color": "GRAY",
"lineType": "BOLD",
"isAsymptote": false
}
},
{
"evalFormula": "x=8",
"displayFormula": "x=8",
"attributes": {
"color": "GRAY",
"lineType": "BOLD",
"isAsymptote": false
}
}
]
},
"fills": [
{
"ranges": [
{
"fst": "-Infinity",
"snd": "2.0"
},
{
"fst": "8.0",
"snd": "Infinity"
}
],
"funcIndices": [],
"funcs": [],
"xIneq": true,
"yIneq": false,
"twoVar": false,
"trueAboveLine": false,
"color": "rgba(171, 181, 235, 0.3)"
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"functionChanges": [
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"origFormulaLatex": [],
"finalFormulaLatex": [],
"plotTitle": "x\\le 2\\lor x\\ge 8",
"paramsLatex": [],
"paramsReplacementsLatex": []
}
],
"localBoundingBox": {
"xMin": -1.5714285714285714,
"xMax": 10,
"yMin": -2,
"yMax": 8
},
"intervals": [
{
"maxd": 2,
"minInclusive": false,
"maxInclusive": true,
"maxFinite": true,
"minFinite": false,
"isAllR": false,
"isPoint": false
},
{
"mind": 8,
"minInclusive": true,
"maxInclusive": false,
"maxFinite": false,
"minFinite": true,
"isAllR": false,
"isPoint": false
}
],
"imageAPI": "x\\le 2\\lor x\\ge 8"
},
"showViewLarger": true
}
},
"meta": {
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}
}
Solution
Solution
+1
Interval Notation
Solution steps
Apply absolute rule: If then or
Graph
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Frequently Asked Questions (FAQ)
What is |x-5|>= 3 ?
The solution to |x-5|>= 3 is x<= 2\lor x>= 8
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