1Hari earns more than Rs. 600 in a day. Taking his earning in one day as \( x \), which inequality is correct: \( x > 600 \), \( x \geq 600 \), \( x < 600 \) or \( x \leq 600 \)?
2In the winter season the temperature of Kathmandu is at least \( -3^\circ\text{C} \). Taking the winter temperature as \( x \), which inequality shows this: \( x > -3^\circ\text{C} \), \( x \geq -3^\circ\text{C} \), \( x < -3^\circ\text{C} \) or \( x \leq -3^\circ\text{C} \)?
3Which of the points \( (0, 0) \), \( (1, 1) \), \( (4, 3) \) and \( (5, 1) \) satisfies the inequality \( 2x - 3y > 5 \)?
4The objective function is \( Z = 3x + 4y \) and the vertices of the feasible region are \( O(0, 0) \), \( A(9, 0) \), \( B(7, 3) \) and \( C(0, 8) \). At which point is Z maximum?
5A graph shows a dashed line through \( (0, 1) \) and \( (2, 0) \), and the side of the line away from the origin is shaded. Which inequality does it show: \( x + y > 2 \), \( x + 2y > 2 \) or \( 2x + y < 2 \)?
6Draw the graph of the inequality \( x + y \geq 3 \).
7Draw the graph of the inequality \( x - y \geq 0 \).
8Draw the graph of the inequality \( 2x + 3y \leq 6 \).
9Draw the graph of the inequality \( 3x + 4y \geq 12 \).
10Draw the graph of the inequality \( x - 2y \geq 3 \).
11Draw the graph of the inequality \( 3x - 2y \geq 4 \).
12Draw the graphs of \( x + 2y \leq 6 \) and \( x \geq 0 \), and find their common solution region.
13Draw the graphs of \( 2x - y \geq 4 \) and \( y \geq 0 \), and find their common solution region.
14Draw the graphs of \( x + y \leq 1 \) and \( x - y \geq 1 \), and find their common solution region.
15Draw the graphs of \( 3x + 4y \leq 12 \), \( 2x - y \leq 0 \), \( x - y \leq 2 \), \( x \geq 0 \) and \( y \geq 0 \), and find their common solution region.
16Draw the graphs of \( 2x + 3y \leq 6 \), \( 3x - y \leq 0 \), \( x \geq 0 \) and \( y \geq 0 \), and find their common solution region.
17Draw the graphs of \( x + y \leq 2 \), \( 2x - 3y \leq 6 \), \( x \geq 0 \) and \( y \leq 2 \), and find their common solution region.
18A graph shows one solid line through the marked points \( (0, 3) \), \( (2, 0) \) and \( (4, -3) \), and the side containing the origin is shaded. Find the inequality shown by the shaded region.
19A graph shows a falling solid line through the marked points \( (-1, 1) \) and \( (2, 0) \), and a rising solid line \( y = x \) through the origin. The region below both lines, to the lower right, is shaded. Find the inequalities shown by the shaded region.
20A graph shows a steep solid line through \( (-1, -1) \) and \( (0, 1) \), and a gentle solid line through \( (-1, -1) \) and \( (2, 0) \). The region to the upper right of both lines is shaded. Find the inequalities shown by the shaded region.
21A graph shows a rising solid line through \( (-1, 0) \) and \( (0, 2) \), a falling solid line through \( (0, 3) \) and \( (4, 0) \), and the X-axis. The small triangle above the X-axis and between the two lines is shaded. Find the inequalities shown by the shaded region.
22Find the maximum and minimum values of \( Z = 3x + 2y \) under the conditions \( x + y \leq 6 \), \( x - y \leq 4 \), \( x \geq 0 \), \( y \geq 0 \).
23Find the maximum and minimum values of \( P(x, y) = 3x + 5y \) under the conditions \( x + y \leq 6 \), \( x - y \geq 2 \), \( x \geq 0 \), \( y \geq 0 \).
24Find the maximum and minimum values of \( Z = 3x + 5y \) under the conditions \( x - 2y \leq 1 \), \( x + y \leq 4 \), \( x \geq 0 \), \( y \geq 0 \).
25Find the maximum and minimum values of \( F = 3x + 2y \) under the conditions \( x - 2y \leq 2 \), \( x + 2y \leq 8 \), \( x \geq 0 \), \( y \geq 0 \).
26Find the maximum and minimum values of \( F(x, y) = 2x + y \) under the conditions \( x + 2y \leq 10 \), \( x + y \geq 1 \), \( y \leq 4 \), \( y \geq 0 \).
27A shaded region has corners \( O(0, 0) \), \( A(3, 0) \), \( B(5, 3) \) and \( C(0, 6) \), bounded by the two axes, a line through A and B, and a line through B and C. Find the four inequalities it shows, and then the maximum and minimum values of \( F(x, y) = 20x + 30y \).
28What is the difference between the objective function and the constraints of a linear programming problem?
29Why is the objective function tested only at the vertices of the feasible region?
30Find the maximum and minimum values of \( Z = 4x + 3y \) under the conditions \( x + y \leq 5 \), \( 2x + y \leq 8 \), \( x \geq 0 \), \( y \geq 0 \).
31On a graph, the crossing point of two boundary lines is hard to read exactly. How can you find it?