the triangle is at least cm, find the minimum length of the shortest side. . A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest.
What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second? [ Hint : If x is the length of the shortest board, then x , ( x + ) and x are the lengths of the second and third piece, respectively. Thus, x + ( x + ) + x ≤ and x ≥ ( x + ) + ]. .
Graphical Solution of Linear Inequalities in Two Variables In earlier section, we have seen that a graph of an inequality in one variable is a visual representation and is a convenient way to represent the solutions of the inequality. Now, we will discuss graph of a linear inequality in two variables. We know that a line divides the Cartesian plane into two parts. Each part is known as a half plane.
A vertical line will divide the plane in left and right half planes and a non-vertical line will divide the plane into lower and upper half planes (Figs. . and . ).
A point in the Cartesian plane will either lie on a line or will lie in either of the half planes I or II. We shall now examine the relationship, if any, of the points in the plane and the inequalities ax + by < c or ax + by > c . Let us consider the line ax + by = c, a ≠ , b ≠ ... ( ) MATHEMATICS Fig .
There are three possibilities namely: ax + by = c ax + by > c (iii) ax + by < c . In case (i), clearly, all points ( x , y ) satisfying (i) lie on the