Exercise 2.10-Additional Questions - Chapter 2 - Algebra - 11th Maths Guide Samacheer Kalvi Solutions

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Additional Questions
Question 1.
$
3 \mathrm{x}+2 \mathrm{y} \leq 12, \mathrm{x} \geq 1, \mathrm{y} \geq 2
$
Solution:
The given inequality is $3 \mathrm{x}+2 \mathrm{y} \leq 12$.
Draw the graph of the line $3 x+2 y \leq 12$
Table of values satisfying the equation
$
3 \mathrm{x}+2 \mathrm{y} \leq 12
$

Putting $(0,0)$ in the given inequation, we have $3 \times 0+2 \times 0 \leq 12$
$\therefore$ Half plane of $3 \mathrm{x}+2 \mathrm{y} \leq 12$ is towards origin
Also the given inequality is $x \geq 1$.
Draw the graph of the line $\mathrm{x}=1$.

Putting $(0,0)$ in the given inequation, we have $0 \geq 1$ which is false.
$\therefore$ Half plane of $\mathrm{x} \geq 1$ is away from origin.
The given inequality is $\mathrm{y} \geq 2$.
Putting $(0,0)$ in the given inequation, we have $0 \geq 2$ which is false.
$\therefore$ Half plane of $y \geq 2$ is away from origin.
Question 2 .
$
\mathrm{x}+\mathrm{y} \geq 4,2 \mathrm{x}-\mathrm{y}>0
$
Solution:
The given inequality is $x+y \geq 4$.
Draw the graph of the line $x+y=4$.

Table of values satisfying the equation $x+y=4$

Putting $(0,0)$ in the given inequation, we have $0+0 \geq 4 \Rightarrow 0 \geq 4$, which is false. $\therefore$ Half plane of $\mathrm{x}+\mathrm{y} \geq 4$ is away from origin.
Also the given inequality is $2 x-y>0$.
Draw the graph of the line $2 \mathrm{x}-\mathrm{y}=0$.
Table of values satisfying the equation $2 \mathrm{x}-\mathrm{y}=0$

Putting $(3,0)$ in the given inequation, we have $2 \times 3-0>0 \Rightarrow 6>0$, which is true. $\therefore$ Half plane of $2 \mathrm{x}-\mathrm{y}>0$ containing $(3,0)$
Question 3.
$x+y \leq 9, y>x, x \geq 0$
Solution:
The given inequality is $\mathrm{x}+\mathrm{y} \leq 9$.
Draw the graph of the line $x+y=9$.

Table of values satisfying the equation
$
x+y=9
$

Putting $(0,0)$ in the given inequation, we have $0+0 \leq 9 \Rightarrow 0 \leq 9$, is towards is origin. $\therefore$ Half plane of $\mathrm{x}+\mathrm{y} \leq 9$ is away from origin.
Also the given inequality is $x-y<0$.
Draw the graph of the line $\mathrm{x}-\mathrm{y}=0$.
Table of values satisfying the equation
$
\mathrm{x}-\mathrm{y}=0
$

Putting $(0,3)$ in the given inequation, we have $0-3-0<0 \Rightarrow-3<0$, which is true. $\therefore$ Half plane of $\mathrm{x}-\mathrm{y}<0$ containing the points $(0,3)$.
Question 4.
$5 x+4 y \leq 20, x \geq 1, y \geq 2$
Solution:
The given inequality is $5 x+4 y \leq 20$.
Draw the graph of the line $5 x+4 y=20$.

Table of values satisfying the equation
$
5 \mathrm{x}+4 \mathrm{y}=20
$

Putting $(0,0)$ in the given inequation, we have $5 \times 0+4 \times 0 \leq 20 \Rightarrow 0 \leq 20$, which is true. $\therefore$ Half plane of $5 \mathrm{x}+4 \mathrm{y} \leq 20$ is away from origin.
Also the given inequality is $x \geq 1$.
Draw the graph of the line $\mathrm{x}=1$.
Putting $(0,0)$ in the given inequation, we have $0 \geq 1$, which is false.
$\therefore$ Half plane of $\mathrm{x} \geq 1$ is $\mathrm{y} \geq 2$.
Draw the graph of the line $\mathrm{y}=2$.
Putting $(0,0)$ in the given inequation, we have $0 \geq 2$, which is false.
$\therefore$ Half plane of $\mathrm{y} \geq 2$ is away from origin.
Question 5 .
$
3 x+4 y \leq 60, x+3 y \leq 30, x \geq 0, y \geq 0
$
Solution:
The given inequality is $3 x+4 y \leq 60$.
Draw the graph of the line $3 x+4 y=60$.
Table of values satisfying the equation
$
3 x+4 y=60
$

Putting $(0,0)$ in the given inequation, we have $3 \times 0+4 \times 0 \leq 60 \Rightarrow 0 \leq 60$, which is true. $\therefore$ Half plane of $3 x+4 y \leq 60$ is towards origin.
Also the given inequality is $\mathrm{x}+3 \mathrm{y} \leq 30$.
Draw the graph of the line $x+3 y=30$.
Table of values satisfying the equation .
$
x+3 y=30
$

Putting $(0,0)$ in the given inequation, we have $0+3 \times 0 \leq 30 \Rightarrow 0 \leq 30$, which is true. $\therefore$ Half plane of $x+3 y \leq 30$ is towards origin.