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LINEAR INEQULATIES, , By, CH SAMUEL, P.G.T MATHEMATICS, J.N.V KOPPAL, Chapter No. :- 6, Class:- 11
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LINEAR INEQULATIES, TOPICS TO BE COVERED, , Introduction, Inequalities, Algebraic solutions of Linear inequalities in one variable and their Graphical Representation, Graphical solution of Linear Inequalities in Two variable, Solution of system of linear inequalities in Two variables
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INTRODUCTION, , In this Chapter, we will study linear inequalities in one and two variables.The study of inequality is very useful in the field of, Science, Mathematics, Statistics, Economics and, Psychology……….etc
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If X is the speed of Car in kmph , , The driver should drive the car at speed less than or equal to 65 kmph, , Symbolically X ≤ 65, If X is height of person in cms, , To join Indian Military, minimum height is 170cms, Symbolically X ≥ 170, Symbols < (less than) and > (greater than) are called strict inequalities, Compare the weight of person A and B, , Symbolically A>B, , Compare the Height of person A and B, , Symbolically A<B, Symbols ≤ (less than or equal to) and ≥ (greater than or equal to) are called slack inequalities
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INEQUALITY, , Two real numbers or two algebraic expressions related by the symbol , <(less than), > (greater than), ≤(less than equal to) or ≥(greater than equal to) form an inequality., But in this chapter we shall confine to the study of linear inequalities in one and two variables only.
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RULES FOR SOLVING AN INEQUALITY, Rule1:Equal numbers may be added to (or substracted from) both sides of an inequality without effecting the sign of inequality, Ex2: 3<10, Subtracting 2 both sides, 3-2<10-2, 1<8, Ex1: 3<10, Adding 5 both sides, 3+5<10+5, 8<15
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RULES FOR SOLVING AN INEQUALITY, Rule 2:Both sides of an inequality can be multiplied (or divided) by the same positive number, it does not effect the sign of inequality, Ex 1: 10>8, Multiply by 5 both sides, 10x5>8x5, , 50>40
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RULES FOR SOLVING AN INEQUALITY, Rule 3:Both sides of an inequality is multiplied or divided by a negative number, then the sign of inequality is reversed, Ex: 6<8, Multiply by -5 both sides, 6x(-5)>8x(-5), , -30>-40
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ALGEBRIC SOLUTION OF LINEAR INEQUALITIES, The values of x which makes the inequality a true statement are called solution of inequality and the set of x values is called solution set of inequality.
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Number Lines Graphical Representation
Page 11 : To represent x<a (or x>a) on a number line, put a circle on the number a and dark line to the left (or right) of the number a.
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To represent x≤a (or x≥a) on a number line, put a dark circle on the number a and dark line to the left (or right) of the number a.
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Ex 1; 3x+8>2 solve i) when x is integer ii) when is x is real number and represent on the number line, sol: we have 3x+8>2, 3x>2-8, 3x>-6, x>-2, i)When x is integer x = -1, 0, 1, 2, ….., , , , ii) When x is Real number, X= (-2, ∞)
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While solving word problems
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Graphical Solution of Linear Inequalities in Two Variables, A vertical line will divide the plane in left and right half planes
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non-vertical line will divide the plane into lower and upper half planes
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Graphical Solution of Linear Inequalities in Two Variables, If an inequality is having ≥ or ≤ symbol, then the points on the line are also included in the solutions of the inequality and the graph of the inequality lies left (below) or right (above) of the graph of the inequality represented by dark line that satisfies an arbitrary point in that part.
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Y’
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Graphical Solution of Linear Inequalities in Two Variables, If an inequality is having < or > symbol, then the points on the line are not included in the solutions of the inequality and the graph of the inequality lies to the left (below) or right (above) of the graph of the corresponding equality represented by dotted line that satisfies an arbitrary point in that part.
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Y’
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Problems, Solve Graphically , x + 2y ≤ 10, x + y ≥1, x – y ≤ 0, x ≥ 0, y ≥ 0
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Q) Solve Graphically x + 2y ≤ 10, x + y ≥1, x – y ≤ 0, x ≥ 0, y ≥ 0, The region containing all the solutions of an inequality is called the solution region .
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Recapitulation, Find x value 2x-4<6 when x is integer, Vertical line divides a plan into ……. Parts, Is (2,3) is a solution of x+y <5?, Express x>2 on number line, Write in algebraic form
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Thank You
