The position of points O (0,0) and P (2, – 2) in the region of graph of inequation 2x − 3y < 5 , will be
Answers
Answer:
x y means x is less than or equal to y.
x y means x is greater than or equal to y.
x < y means x is less than y.
x > y means x is greater than y.
The last two inequalities are called strict inequalities. Our focus will be on the
nonstrict inequalities.
Algebra of Inequalities
Suppose
x + 3 < 8.
Addition works like for equations:
x + 6 < 11 (added 3 to each side).
Subtraction works like for equations:
x + 2 < 7 (subtracted 4 from each side).
Multiplication and division by positive numbers work like for equations:
2x + 12 < 22 =) x + 6 < 11
(each side is divided by 2 or multiplied by
1
2
).
59
60 4. LINEAR PROGRAMMING
Multiplication and division by negative numbers changes the direction of the
inequality sign:
2x + 12 < 22 =) x 6 > 11
(each side is divided by -2 or multiplied by 1
2
).
Example. For 3x 4y and 24 there are 3 possibilities:
3x 4y = 24 3x 4y < 24 3x 4y > 24
4y = 3x + 24 4y < 3x + 24 4y > 3x + 24
y = 3
4x 6 y > 3
4x 6 y < 3
4x 6
The three solution sets above are disjoint (do not intersect or overlap), and their
graphs fill up the plane. We are familiar with the graph of the linear equation.
The graph of one inequality is all the points on one side of the line, the graph
of the other all the points on the other side of the line.
To determine which side for an inequality, choose a test point not on the line
(such as (0, 0) if the line does not pass through the origin). Substitute this
point into the linear inequality. For a true statement, the solution region is the
side of the line that the test point is on; for a false statement, it is the other
side.
1. GRAPHING LINEAR INEQUALITIES 61
3x 4y < 24 3x 4y > 24
0 0 < 24 (true) 0 0 > 24 (false)
—————— ——————
y > 3
4x 6 y < 3
4x 6
0 > 0 6 (true) 0 < 0 6 (false)
Statement is true, Statement is false,
so take side of line (0, 0) is on. so take side opposite (0, 0).
The dashed lined indicates that points on the line are not part of the solution.
For 3x 4y 24 or 3x 4y 24, we would use a solid line to indicate that
the boundary line is part of the solution.
Note.
(1) For y mx + b, the solution region is the line with the points above it.
(2) For y mx + b, the solution region is the line with t
Step-by-step explanation:
Answer:
The position of points O (0,0) and P (2, – 2) in the region of the graph of inequation 2x − 3y < 5 can be determined by substituting the coordinates for x and y into the inequality.
When x = 0 and y = 0, 2 (0) − 3 (0) < 5, which is true.
This means that 0 multiplied by 2 minus 0 multiplied by 3 is less than 5. Therefore, point O (0, 0) is in the region of the graph of inequation 2x − 3y < 5.
When x = 2 and y = – 2, 2 (2) − 3 (−2) < 5, which is also true.
This means that 2 multiplied by 2 minus -2 multiplied by 3 is less than 5. Therefore, point P (2, – 2) is in the region of the graph of inequation 2x − 3y < 5.
It is important to remember that when dealing with inequalities that involve multiplication or division, both sides of the inequality must be multiplied or divided by the same number. This helps to ensure that the inequality is not changed and that the points can be accurately located.
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