Linear Inequalities
Board: CBSE |Class:Class11
ComprehensivestudynotesforLinearInequalitiesbyAjayYadav(MathKingofKatargam).Mastereveryconceptwithclearexplanations,solvedexamples,andpracticeproblems.
Aninequalityisastatementwith<,>,≤,or≥.Rules:Adding/subtractingsamenumberpreservesinequality.Multiplying/dividingbypositivenumberpreservesitbynegativenumberreversesit.
Solvelikeequationsbutreverseinequalitysignwhenmultiplying/dividingbynegative.Example:-2x>6⇒x<-3.
Inequalitiesconnectedby”and”(intersection)or”or”(union).Solveeachpartseparately.For”and”:takecommonsolution.For”or”:combinesolutions.
Inequalitiesintwovariables:drawtheboundarylineax+by=c(solidif≤/≥,dashedif</>).Shadetheregionthatsatisfiestheinequality(testby(0,0)).
Thesolutionregionistheintersectionofallindividualinequalityregions.Feasibleregion=commonshadedarea.Cornerpointsarewhereboundarylinesintersect.
| Addition | a<b⇒a+c<b+c |
| Multiplication(positive) | a<b,c>0⇒ac<bc |
| Multiplication(negative) | a<b,c<0⇒ac>bc |
| Compound”and” | Solution=Intersectionofboth |
| Compound”or” | Solution=Unionofboth |
Example1:Solve:2x-3<7.
Solution:2x<10⇒x<5.Solution:(-∞,5).
Example2:Solve:3x+2≥5x-4.
Solution:3x-5x≥-4-2⇒-2x≥-6⇒x≤3.Solution:(-∞,3].
Example3:Solvegraphically:x+y≤5,x≥1,y≥2.
Solution:Drawlinesx+y=5,x=1,y=2.Shadedregionistrianglewithvertices(1,2),(3,2),(1,4).Feasibleregionisbounded.
- Solve:5x-3≥3x+1.
- Solve:-3≤2x-1<5.
- Solvegraphically:2x+y≥6,x+2y≤8,x≥0,y≥0.
- The sumoftwonumbersisatmost20andeachisatleast5.Writeasinequalities.
- Solve:|x – 2| < 3.
Download PDF
Click here to download the PDF notes.
Video Lessons
Watch video explanations on our Videos page.