Answer :
Let be "x" the number of nickels Evelyn has and "y" the number of pennies she has.
According to the information given in the exercise, she has a maximum of 20 coins. Then, you can set up the following inequality to represent it:
[tex]x+y\le20[/tex]Because "a maximum of 20 coins" indicates that the total number of coins is less than or equal to 20.
You also know that that money is worth a minimum of $0.40. Then, you can set up this inequality:
[tex]0.05x+0.01y\ge0.40[/tex]Since the first inequality is:
[tex]x+y\le20[/tex]You need to solve for "y" in order to rewrite it:
[tex]y\le-x+20[/tex]Knowing that the second inequality is:
[tex]0.05x+0.01y\ge0.40[/tex]You can solve for "y" in order to rewrite it:
[tex]\begin{gathered} 0.01y\ge-0.05x+0.40 \\ \\ y\ge\frac{-0.05x}{0.01}+\frac{0.40}{0.01} \\ \\ y\ge-5x+40 \end{gathered}[/tex]Therefore, the System of Inequalities is:
[tex]\begin{cases}y\le-x+20 \\ y\ge-5x+40\end{cases}[/tex]The Slope-Intercept Form of the equation of a line is:
[tex]y=mx+b[/tex]Where "m" is the slope and "b" is the y-intercept.
- Notice that the first line of the system is:
[tex]y=-x+20[/tex]You can identify that:
[tex]b_1=20[/tex]You can find the x-intercept by substituting the following value of "y" into the equation and solving for "x":
[tex]y=0[/tex]Then, you get:
[tex]\begin{gathered} 0=-x+20 \\ x=20 \end{gathered}[/tex]Therefore, you know that the first line passes through these points:
[tex](20,0);(0,20)[/tex]- Since the second line is:
[tex]y=-5x+40[/tex]You can determine that:
[tex]b_2=40[/tex]To find the x-intercept, apply the same procedure used for the first line:
[tex]\begin{gathered} 0=-5x+40 \\ -40=-5x \\ \\ \frac{-40}{-5}=x \\ \\ x=8 \end{gathered}[/tex]Then, the line passes through these points:
[tex](8,0);(0,40)[/tex]- Notice that the symbol of the first inequality is:
[tex]\le[/tex]That indicates that the first line is solid and the shaded region must be below the line.
- The symbol of the second inequality is:
[tex]\ge[/tex]This indicates that the line is solid and the shaded region must be above the line.
Knowing the explained above, you can graph the System of Inequalities:
By definition, the solution of the System of Inequalities is the intersection region.
Then, in order to determine one possible solution, you can choose a point in the intersection region. This can be (the solution contains this point):
[tex](10,8)[/tex]Therefore, answers are:
-Inequality 1:
[tex]y\le-x+20[/tex]- Inequality 2:
[tex]y\ge-5x+40[/tex]- Graph:
- The solution contains this point:
[tex](10,8)[/tex]
