## Solve Lesson 25 Page 32 SBT Math 10 – Kite>

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.