Chapter 10, Problem 103RE

To determine

To shade: the given solution set

Answer to Problem 103RE

  

Explanation of Solution

Given:

  $\{\begin{array}{l}y\ge x-1\hfill \\ x^2+y^2\le 1\hfill \end{array}$

  

Calculation:

Consider the system of equations,

  $\{\begin{array}{l}x+y\ge 2\hfill \\ y-x\le 2\hfill \\ x\le 3\hfill \end{array}$

since the system of inequalities includes the equal signs.

So, draw solid lines that indicate that the points on the lines are included in the solution set.

The Graph of $x+y=2$ is a solid line as shown by red color,

The Graph of $y-x=2$ is a solid line as shown by blue color.

The Graph of $x=3$ is a solid line as shown by green color.

  

Now choose a test point to see which side of the lines satisfies the inequality.

Test the ordered pair (0,0) for $x+y\ge 2$ as,

  $x+y\ge 2$

  $0+0{\ge }^{7}2$

  $0\ge 3$ False

That means (0,0) does not belong to the solution set of $x+y\ge 2$

So shade the portion of plane towards (0,0) i.e. above the line $x+y=2$

  

Now choose a test point to see which side of the lines satisfies the inequality.

Test the ordered pair (0,0) for $y-x\le 2$ as,

  $y-x\le 2$

  $0-0\le ?$

  $0\le 3$ TRUE

That means (0,0) does not belong to the solution set of $y-x\le 2$

So shade the portion of plane towards (0,0) i.e. below the line $y-x=2$

  

Next

Pick (0,0) as a test point to determine which half-plane to shade and substitute (0,0) in $x\le 3$

  $x\le 3$

  $0\le 3$

  $0\le 3$ true

That means (0,0) belongs to the solution set of $x\le 3$

So shade the portion of plane containing (0,0) i.e. below the line $x=3$

  

Therefore the final graph is as shown below.

That is, the graph of the solution set of the system of inequalities is given by the intersection of the graphs of the solution sets of $\{\begin{array}{l}x+y\ge 2\hfill \\ y-x\le 2\hfill \\ x\le 3\hfill \end{array}$

  

Conclusion:

Therefore, the graph is shaded.