Chapter 10.9, Problem 2E

(a)

To determine

To sketch: The shaded solution part of the graph of given system of inequalities.

Explanation of Solution

The given systems of inequalities are,

$x-y\ge 0$ (1)

$x+y\ge 2$ (2)

Choose any point in the x-y plane and check whether that point satisfies the inequalities or not, if yes then shade that portion of the graph.

Check for another point $\left(3,2\right)$,

Substitute 3 for x and 2 for y in equation (1).

$\begin{array}{c}3-2\ge 0\\ 1\ge 0\end{array}$

This point satisfies the inequality,

Therefore, the shaded portion of the graph is on the right side of the graph $x-y=0$

Plot the graph of the inequality according to the above solution.

Figure (1)

Figure (1) shows the graph of the inequality $x-y\ge 0$.

Substitute 3 for x and 2 for y in equation (1).

$\begin{array}{c}3+2\ge 2\\ 5\ge 2\end{array}$

This point also satisfies the inequality,

Therefore, the shaded portion is on the right side of the graph. and away from the origin.

Plot the graph according to the above solution.

Figure (2)

The Figure (2) shows the graph of the inequality $x+y\ge 2$.

Draw both the inequalities on the same plane and shade the common solution of the given system of inequalities,

Figure (3)

The Figure (a) shows the graph of the inequality. $x-y\ge 0$, $x+y\ge 2$.

(b)

To determine

To sketch: The shaded solution part of the graph of given system of inequalities.

Explanation of Solution

The given systems of inequalities are,

$x-y\le 0$ (1)

$x+y\le 2$ (2)

Choose any point in the x-y plane and check whether that point satisfies the inequalities or not, if yes then shade that portion of the graph

Now, check for another point $\left(1,2\right)$,

Substitute 1 for x and 2 for y in both the equations.

From equation (1)

$\begin{array}{c}1-2\le 0\\ -1\le 0\end{array}$

This point satisfies the inequality,

Therefore, the shaded portion of the graph is on the left side of the graph $x-y=0$

Plot the graph of the inequality according to the above solution.

Figure (4)

The Figure (4) shows the graph of the inequality $x-y\le 0$.

From, equation (2)

$\begin{array}{c}1+2\le 2\\ 3\le 2\end{array}$

This point does not satisfies the inequality,

Therefore, the shaded portion is on the left side of the graph. and towards the origin.

Plot the graph of the inequality according to the above solution.

Figure (5)

The Figure (5) shows the graph of the inequality $x+y\le 2$.

Plot both the inequalities on the same plane and shade the common solution of the given system of inequalities,

Figure (6)

Figure (6) shows the graph of the given inequalities $x-y\le 0$, $x+y\le 2$.

(c)

To determine

To sketch: The shaded solution part of the graph of given system of inequalities.

Explanation of Solution

The given systems of inequalities are,

$x-y\ge 0$ (1)

$x+y\le 2$ (2)

Choose any point in the x-y plane and check whether that point satisfies the inequalities or not, if yes then shade that portion of the graph

Check for another point $\left(3,2\right)$,

Substitute 3 for x and 2 for y in both the equations.

From equation (1)

$\begin{array}{c}3-2\ge 0\\ 1\ge 0\end{array}$

This point satisfies the inequality,

Therefore, the shaded portion of the graph is on the right side of the graph $x-y=0$

Plot the graph of the inequality according to the above solution.

Figure (1)

Figure (1) shows the graph of the inequality $x-y\ge 0$.

From equation (2)

$\begin{array}{c}3+2\le 2\\ 5\le 2\end{array}$

This point does not satisfies the inequality,

Therefore, the shaded portion is on the left side of the graph and towards the origin.

Therefore, the shaded portion is on the left side of the graph. and towards the origin.

Plot the graph of the inequality according to the above solution.

Figure (5)

The Figure (5) shows the graph of the inequality $x+y\le 2$.

Plot both the inequalities on the same plane and Shade the common solution of the given system of inequalities.

Figure (7)

Figure (7) shows the graph of the given inequalities $x-y\ge 0$, and $x+y\le 2$.

(d)

To determine

To sketch: The shaded solution part of the graph of given system of inequalities.

Explanation of Solution

The given systems of inequalities are,

$x-y\ge 0$ (1)

$x+y\le 2$ (2)

Choose any point in the x-y plane and check whether that point satisfies the inequalities or not, if yes then shade that portion of the graph

Now, check for another point $\left(1,2\right)$,

Substitute 1 for x and 2 for y in both the equations.

From equation (1)

$\begin{array}{c}x-y\le 0\\ 1-2\le 0\\ -1\le 0\end{array}$

This point satisfies the inequality,

Therefore, the shaded portion of the graph is on the left side of the graph $x-y=0$

Plot the graph of the inequality according to the above solution.

Figure (4)

The Figure (4) shows the graph of the inequality $x-y\le 0$.

From, equation (2)

$\begin{array}{c}x+y\ge 2\\ 1+2\ge 2\\ 3\ge 2\end{array}$

This point satisfies the inequality,

Therefore, the shaded portion is on the right side of the graph. and away from the origin.

Plot the graph of the inequality according to the above solution.

Figure (5)

The Figure (5) shows the graph of the inequality $x+y\le 2$.

Plot the graph of both the inequalities on the same plane and shade the common solution of the given system of inequalities,

Figure (8)

Figure (8) shows the graph of the given inequalities