introduction today we will learn linear inequality can anyone tell me what is inequality inequality involves less than or more than sign very good teacher then what is linear inequality linear inequalities are the statements which involves inequality sign in this lesson you will learn how to solve linear inequalities problems objective at the end of this lesson you will be able to State the linear inequality find the inequalities find the solution of linear inequalities in one variable find the
solution of linear inequalities in two variables represent the linear inequalities graphically inequalities linear inequalities are the statement that involve the sign of inequality these are less than greater than less than or equal greater than or equal not equal let's have some examples for better understanding example one to enter a junior writing competition you must be under 12 years old so the inequality for the statement is age is less than 12. under 12 means less than 12 and does not in
clude equal to 12 therefore the correct symbols is less than it is a linear inequality of one variable example two Suresh went to Market to buy hanky and socks with rupees 200 the cost of one hanky is rupees 25 the cost of one pair of socks is rupees 30 if x denotes the number of hanky and Y denotes the number of socks pairs which Suresh buy then we can represent the above statement in inequality form as 25x plus 30 y less than or equal to 200 it is a linear inequality of two variables it can be
written in two statements 25x plus 35 is equal to 200 25x plus 35 less than 200 name it as a statement a is an equality in this case example the length of a rectangle is greater than 5 centimeter and its width is greater than 4 centimeter write the inequality for the area of the rectangle solution the area of a rectangle is found by multiplying the length by the width if the length was equal to 5 centimeter and the width was equal to 4 centimeter then the area would be 5 centimeter into four ce
ntimeter is equal to 20 centimeter square but the length is greater than 5 centimeter and the width is greater than 4 centimeter so the area is greater than 20 centimeter Square algebraic solution of linear inequalities in one variable and their graphical representation the solution of an inequality of one variable is the value of the variable which makes it a true statement tools to solve a linear inequality in one variable number one equal numbers may be added to or subtracted from both sides
of an inequality without affecting the sign of inequality number two both sides of an inequality can be multiplied or divided by the same positive number but when both sides are multiplied or divided by a negative number then the sign of inequality is reversed number three any term on any side of inequality can be transferred to the other side with its sign change without affecting the sign of inequality example solve 4X minus 19 less than 15x plus 14 show the graph of the solution on the number
line solution transposing 15x to LHS and 19 to rhs we have 4X minus 15x is less than 14 plus 19 or minus 11x is less than 33 minus X is less than 3 x is greater than -3 this inequality is shown on a number line first we draw an open circle because X is not equal to -3 and an arrow to the right because we want values greater than four number line example Ram needs a minimum of 360 marks in four tests in a mathematics course to obtain a grade on his first three tests he scored 88 96 79 marks what
should his score be in the fourth test so that he can make a grade solution let Ram scores x marks in the fourth test then the sum of the Rams test scores should be greater than or equal to 360. that is 88 plus 96 plus 79 plus X greater than or equal to 360. it implies 263 plus X greater than or equal to 360. it implies X greater than or equal 97 Ram should score 97 or greater than 97 in the fourth test to obtain a grade graphical solution of linear inequalities in two variables let ax plus b y
plus C star 0 be a linear inequality in variables X and Y where a B and C are arbitrary real numbers and star is the inequality sign the linear equation ax plus b y plus C is equal to 0 plot is a straight line in X Y plane below are the examples of straight line in X Y plane which divides X Y plane into two half the graph of linear inequality is a half plane which is determined by the linear equation the straight line divides the X Y plane in two parts that is half planes and exactly one of the
se would be the graph of the inequality to decide as to which of the half plane is the solution of linear inequality we take an arbitrary Point p h comma k there are two cases case one p satisfies linear inequality if the point p h comma K satisfies the linear inequality then the required graph of the given inequality is that half plane which contains the point pH comma K case 2 p does not satisfy linear inequality if the point p h comma K does not satisfy the linear inequality then the required
graph of the given inequality is that half plane which does not contain the point p h comma K the set of points on the half plane represented by linear inequality is called the solution set of the given inequality if the inequality involve sine less than or equal or greater than or equal then the points satisfying the straight line ax plus b y plus C is equal to 0 R also included in the solution set of inequality and on the other hand a if inequality involve sine less than or greater than then
the points satisfying the straight line ax plus b y plus C is equal to 0 are not included in the solution set of inequality node graph of ax plus b y plus C is equal to 0 is a dashed line if equality is not included in the original statement or as a solid line if inequality is included example draw the graph of x plus 2y less than or equal to 10 solution the given inequality is X Plus 2y less than or equal to 10. the corresponding linear equation is X Plus 2y is equal to 10. name it as 1. now we
shall draw the line 1 when X is equal to 0 Line 1 implies 0 plus 2y is equal to 10 Y is equal to 5 so we plot the number 0 comma 5 when Y is equal to 0 Line 1 implies X plus 2 into 0 is equal to 10 x is equal to 10 so we plot the point 10 comma 0 the line joining 0 comma 5 and 10 comma 0 divide the X Y plane in two half planes now we select a point say 0 comma 0 which is one of the half planes and determine if this point satisfies the given inequality 0 plus 2 into 0 less than or equal to 10 wh
ich is true hence half plane 1 is the solution region of the given inequality example solve the following system of linear inequalities graphically and find the corner points 2x plus Y is less than or equal to 22. X Plus Y is less than or equal to 13. 2x plus 5y is less than or equal to 50. X is greater than or equal to zero Y is greater than or equal to zero solution the inequalities X greater than or equal to 0 and y greater than or equal to 0 called non-negative restrictions the solution regi
on lies in the first quadrant and we can restrict our attention to that portion of the brain first we graph the lines as follows 2x plus Y is equal to 22 at X is equal to 0 Y is equal to 22 so the point is 0 comma 22 at Y is equal to 0 x is equal to 11 so that point is 11 comma 0 X Plus Y is equal to 13 and X is equal to 0 Y is equal to 13 so the point is 0 comma 13 at Y is equal to 0 x is equal to 13 so the point is 13 comma 0. 2x plus 5y is equal to 50 at X is equal to 0 Y is equal to 10 so th
e point is 0 comma 10 at Y is equal to 0 x is equal to 25 so the point is 25 comma 0 next choosing 0 comma 0 as a test point we see that the graph of each of the first three inequalities in the system consists of its corresponding line and the half plane line below the line as indicated by the arrows thus the solution region of the system consists of the points in the first quadrant that simultaneously lie on or below all three of these lines let's find out Corner points the corner points can be
extracted from the figure as 0 comma zero zero comma 10 11 comma 0 and the other two points can be determined by solving the equations 2x Plus 5y is equal to 50 X Plus Y is equal to 13. by solving the equations we get 5 comma 8. 2x plus Y is equal to 22 X Plus Y is equal to 13 by solving the equations we get 9 comma four so the corner points are 0 comma zero zero comma 10 11 comma zero five comma 8 9 comma 4. note that the lines 2x plus 5y is equal to 50 and 2x plus Y is equal to 22 also inters
ect but the intersection point is not part of the solution region and hence is not a corner point did you know William Henry young was an English mathematician has important contribution in inequalities the inequality signs were first used by the English mathematician Thomas Herriot summary let us summarize what we have learned linear inequalities are the statement that involve the sign of inequality these are less than greater than less than or equal greater than or equal not equal rules to sol
ve a linear inequality in one variable a equal numbers may be added to or subtracted from both sides of an inequality without affecting the sign of inequality B both sides of an inequality can be multiplied or divided by the same positive number but when both sides are multiplied or divided by a negative number then the sign of inequality is reversed C any term on any side of inequality can be transferred to the other side with its sign changed without affecting the sign of inequality graphical
solution of linear inequalities in two variables if the inequality involved sine less than or equal or greater than or equal then the points satisfying the straight line ax plus b y plus C is equal to 0 are also included in the solution set of inequality and on the other hand if inequality involve sine less than or greater than then the points satisfying the straight line ax plus b y plus C is equal to 0 are not included in the solution set of inequality graph of inequality is a dashed line if e
quality is not included in the original statement or as a solid line if equality is included
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