Chapter 6.6, Problem 30PPE

To determine

To find:determinethe error in the given graph drawn by a student.

Explanation of Solution

Given information:

Given graph is as follows

Conditions given in the graph are

  $x\ge -2$ … (i)

  $y>2$ … (ii)

  $\frac{1}{2}x+y\ge 0$ … (iii)

The graph plotted for equation (i), (ii) and (iii) are correct; however the shaded portion is incorrect.

Shaded portion is to be drawn between two lines such that it satisfies all the condition.

According to equation (i) solution should contain the area such that $x\ge -2$ .

This is done by drawing a line at $x=-2$ and since it says x should be greater than $-2$ , are only to right of this line is considered as solution.

According to equation (ii), $y>2$ , here it must be noted that y is not equal to $2$ but greater than $2$ so rather than drawing a line, only a dotted line is made for reference.

As per this condition the solution should have area such that at any point y co-ordinate of that point is greater than $2$ . That means only the upper side of the reference line should be part of solution.

In the graph plotted by student, condition (ii) is not fulfilled.

According to eqaution (iii), $\frac{1}{2}x+y\ge 0$ , which is an equation of line.

It can be solved by converting this equation into slope-intercept equation of line which is

  $\begin{array}{l}\frac{1}{2}x+y=0\\ y=-\frac{1}{2}x\end{array}$

This becomes an equation of a line which a slope of $-\frac{1}{2}$ .

In the graph plotted by the student the line has positive slope which is incorrect.

Therefore conditon (iii) is also not fulfilled.

The correctly plotted graph would be

  

And the correctly shaded graph as solution will be

  

Conclusion:

Therefore, the portion shaded as gray should be a part of solution.