Use the image to answer the question.

Question

A coordinate plane shows a graphed dashed line. The x-axis and y-axis both range from negative 10 to 10 in one unit increments. No points are plotted. The graphed line intercepts grid lines at coordinates left parenthesis negative 10 comma negative 2 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 10 comma 2 right parenthesis. The area above the dashed line is shaded.

Determine the whole number unknowns for the solution set of the given graph of an inequality in two variables.

Answer 1

To determine the solution set of the given graph of the dashed line and the area that is shaded above the line, we’ll first analyze the information provided.

1. Identify the Equation of the Line: The points provided where the line intercepts the grid lines are:

   • $(-10,-2)$
   • $(0,0)$
   • $(10,2)$

   From the two points $(0,0)$ and $(10,2)$, we can derive the equation of the line.

   The slope (m) of the line can be calculated as: $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{2-0}{10-0}=\frac{2}{10}=\frac{1}{5}$$

   Using the point-slope form of a linear equation $y-y_1=m(x-x_1)$ with the point $(0,0)$, we get: $$y-0=\frac{1}{5}(x-0)⟹y=\frac{1}{5}x$$

2. Determine the Inequality: Since the area above the line is shaded, this indicates that the inequality represented by the line is: $$y>\frac{1}{5}x$$

3. Finding the Whole Number Solutions: We need to find the whole number solutions $(x,y)$ that satisfy the inequality $y>\frac{1}{5}x$.

   To find suitable whole number pairs, we can test integer values of $x$ and calculate $y$ accordingly:

   For $x=-10$: $$y>\frac{1}{5}(-10)=-2\Rightarrow y\ge -1\text{ (as y must be a whole number)}$$ Solutions: $(-10,-1),(-10,0),(-10,1),(-10,2),...,(-10,10)$

   For $x=-9$: $$y>\frac{1}{5}(-9)=-1.8\Rightarrow y\ge -1$$ Solutions: $(-9,-1),(-9,0),(-9,1),...,(-9,10)$

   Continue for $x=-8$ to $x=10$:

   For $x=-8$: $$y>\frac{1}{5}(-8)=-1.6\Rightarrow y\ge -1$$ Solutions: $(-8,-1),(-8,0),(-8,1),...,(-8,10)$

   For $x=-7$: $$y>\frac{1}{5}(-7)=-1.4\Rightarrow y\ge -1$$ Solutions: $(-7,-1),(-7,0),(-7,1),...,(-7,10)$

   For $x=-6$: $$y>\frac{1}{5}(-6)=-1.2\Rightarrow y\ge -1$$ Solutions: $(-6,-1),(-6,0),(-6,1),...,(-6,10)$

   For $x=-5$: $$y>\frac{1}{5}(-5)=-1\Rightarrow y\ge 0$$ Solutions: $(-5,0),(-5,1),...,(-5,10)$

   Continue this process until $x=10$:

   For $x=10$: $$y>\frac{1}{5}(10)=2\Rightarrow y\ge 3$$ Solutions: $(10,3),(10,4),(10,5),...,(10,10)$

4. Summarization of Whole Number Solutions: So, the whole number solutions $(x,y)$ in the area above the line includes:

• For $x$ values from -10 to 10.
• Whole $y$ values such that $y>\frac{1}{5}x$.

In conclusion, every integer point in the shaded region above the dashed line forms a part of the solution set for the given inequality $y>\frac{1}{5}x$. The exact values of $y$ depend on the specific $x$ as described above.