**Topic**

Express the solution domain of the following inequalities:

a) \(3x > 2\) b) \(2y \le – 5\) c) \(2x – y \ge 1\) d) \(3x – 2y < 5\)

**Solution method – See details**

Representation of the root domain of bpt \(ax + by < c\)

Step 1: Draw a line \(d:ax + by = c\).

Step 2: Get the point \(M\left( {{x_o};{y_o}} \right)\) not on d (we usually take the origin O if \(c \ne 0\)). Calculate \(a{x_o} + b{y_o}\) and compare with c

Step 3: Conclusion

If \(a{x_o} + b{y_o} < c\) then the half-plane (excluding line d) containing the point M is the solution domain of the inequality \(ax + by < c\)

If \(a{x_o} + b{y_o} > c\) then the half-plane (excluding d) containing no point M is the solution domain of the inequality \(ax + by > c\)

**Detailed explanation**

a) Draw a line: \(3x = 2\)

Considering the point O(0; 0) we have 3.0 = 0 < 2, so O(0;0) is not in the solution domain of bpt \(3x > 2\).

The solution domain of the inequality \(3x > 2\) is the half-plane edge a, which does not contain the point O.

b) Draw a line b: 2y = – 5

Considering O(0; 0) we have 2.0 = 0 > – 5.

=> O(0; 0) is not in the solution domain of bpt \(2y \le – 5\)

Therefore, the solution domain of the inequality \(2y \le – 5\) is a half-plane of shore b, which does not contain the point O.

c) Draw a line c: 2x – y = 1

Considering the point O(0; 0) we have 2.0 – 0 = 0 < 1.

=> O(0; 0) is not in the solution domain of bpt \(2x – y \ge 1\)

Therefore, the solution domain of the inequality \(2x – y \ge 1\) is a half-plane of edge c, which does not contain the point O.

d) Draw a line d: 3x – 2y = 5

Considering the point O(0; 0) we have 3.0 – 2.0 = 0 < 5.

=> O(0; 0) belongs to the solution domain of bpt \(3x – 2y < 5\)

Therefore, the solution domain of the inequality \(3x – 2y < 5\) is a half-plane of shore d, containing the point O.