{
"query": {
"display": "$$\\left|x-7\\right|\\le\\:3$$",
"symbolab_question": "EQUATION#\\left|x-7\\right|\\le 3"
},
"solution": {
"level": "PERFORMED",
"subject": "Algebra",
"topic": "Inequalities",
"subTopic": "Absolute",
"default": "4\\le x\\le 10",
"interval": "[4,10]",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\left|x-7\\right|\\le\\:3{\\quad:\\quad}4\\le\\:x\\le\\:10$$",
"input": "\\left|x-7\\right|\\le\\:3",
"steps": [
{
"type": "step",
"primary": "Apply absolute rule: If $$|u|\\:\\leq\\:a,\\:a\\:>\\:0\\:$$then $$-a\\:\\leq\\:u\\:\\leq\\:a$$",
"result": "-3\\le\\:x-7\\le\\:3"
},
{
"type": "interim",
"title": "$$x-7\\ge\\:-3\\land\\:x-7\\le\\:3$$",
"input": "x-7\\ge\\:-3\\land\\:x-7\\le\\:3",
"result": "x\\ge\\:4\\land\\:x\\le\\:10",
"steps": [
{
"type": "interim",
"title": "$$x-7\\ge\\:-3{\\quad:\\quad}x\\ge\\:4$$",
"input": "x-7\\ge\\:-3",
"steps": [
{
"type": "interim",
"title": "Move $$7\\:$$to the right side",
"input": "x-7\\ge\\:-3",
"result": "x\\ge\\:4",
"steps": [
{
"type": "step",
"primary": "Add $$7$$ to both sides",
"result": "x-7+7\\ge\\:-3+7"
},
{
"type": "step",
"primary": "Simplify",
"result": "x\\ge\\:4"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Inequalities",
"interimType": "Inequalities"
}
},
{
"type": "interim",
"title": "$$x-7\\le\\:3{\\quad:\\quad}x\\le\\:10$$",
"input": "x-7\\le\\:3",
"steps": [
{
"type": "interim",
"title": "Move $$7\\:$$to the right side",
"input": "x-7\\le\\:3",
"result": "x\\le\\:10",
"steps": [
{
"type": "step",
"primary": "Add $$7$$ to both sides",
"result": "x-7+7\\le\\:3+7"
},
{
"type": "step",
"primary": "Simplify",
"result": "x\\le\\:10"
}
],
"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\\ge\\:4\\land\\:x\\le\\:10"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "interim",
"title": "Merge Overlapping Intervals",
"input": "x\\ge\\:4\\land\\:x\\le\\:10",
"result": "4\\le\\:x\\le\\:10",
"steps": [
{
"type": "step",
"primary": "The intersection of two intervals is the set of numbers which are in both intervals<br/>$$x\\ge\\:4\\quad$$and$$\\quad\\:x\\le\\:10$$",
"image": "/images/interval?expression=%28y_%7B0%7D%5Cge+4%29%5Cland+%28y_%7B0%7D%5Cle+10%29",
"result": "4\\le\\:x\\le\\:10"
}
],
"meta": {
"interimType": "Merge Overlapping Intervals 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQvFNW32lPq3eSYZSCJVIhaNm9nslQdQAqfvL1PekEGbtFKk3fejFkyiOiq9iG9IkAnElIGQ3VI0QtHiJqvLzKFT6qDGX6RVNIyTGGHqMe0Gi9RQa+VFzMcO8uN2y3ZuWk"
}
}
],
"meta": {
"solvingClass": "Inequalities"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"funcsToDraw": {
"funcs": [
{
"evalFormula": "x=4",
"displayFormula": "x=4",
"attributes": {
"color": "GRAY",
"lineType": "BOLD",
"isAsymptote": false
}
},
{
"evalFormula": "x=10",
"displayFormula": "x=10",
"attributes": {
"color": "GRAY",
"lineType": "BOLD",
"isAsymptote": false
}
}
]
},
"fills": [
{
"ranges": [
{
"fst": "4.0",
"snd": "10.0"
}
],
"funcIndices": [],
"funcs": [],
"xIneq": true,
"yIneq": false,
"twoVar": false,
"trueAboveLine": false,
"color": "rgba(171, 181, 235, 0.3)"
}
],
"functionChanges": [
{
"origFormulaLatex": [],
"finalFormulaLatex": [],
"plotTitle": "4\\le x\\le 10",
"paramsLatex": [],
"paramsReplacementsLatex": []
}
],
"localBoundingBox": {
"xMin": -1.7142857142857142,
"xMax": 12,
"yMin": -2,
"yMax": 8
},
"intervals": [
{
"mind": 4,
"maxd": 10,
"minInclusive": true,
"maxInclusive": true,
"maxFinite": true,
"minFinite": true,
"isAllR": false,
"isPoint": false
}
],
"imageAPI": "4\\le x\\le 10"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solution
Solution
+1
Interval Notation
Solution steps
Apply absolute rule: If then
Merge Overlapping Intervals
Graph
Popular Examples
Frequently Asked Questions (FAQ)
What is |x-7|<= 3 ?
The solution to |x-7|<= 3 is 4<= x<= 10
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