We are given that x is the number of nickels and y the number of dimes. Since there must be at maximum 28 of them, we can express this mathematically like this:
[tex]x+y\le28[/tex]
We are also told that they combine must be worth a minimum of $2, this means mathematically:
[tex]0.05x+0.1y\ge2[/tex]
Now we are told that the number of nickels is less than 8 and the number of dimes is more than 22, this can be expressed mathematically as:
[tex]\begin{gathered} x\le8 \\ y\ge22 \end{gathered}[/tex]
Therefore we have the following system of inequalities
[tex]\begin{gathered} x+y\le28,\text{ (1)} \\ 0.05x+0.1y>=2,\text{ (2)} \\ x\le8,\text{ (3)} \\ y\ge22,\text{ (4)} \end{gathered}[/tex]
The graph this inequation is the following:
The possible solutions to the inequations are located where all the colors intercept.
For example, we can take the following point:
[tex](x,y)=(5,22)[/tex]
For inequality (1)
[tex]\begin{gathered} x+y\le28 \\ 5+22\le28 \\ 27\le28 \end{gathered}[/tex]
For inequality (2)
[tex]\begin{gathered} 0.05x+0.1y\ge2 \\ 0.05(5)+0.1(22)\ge2 \\ 2.45\ge2 \end{gathered}[/tex]
Since 5<8 and 22=22 this is a solution to the inequality.
The inequalities can be rewritten as:
[tex]\begin{gathered} y\le28-x,\text{ (1)} \\ y\le\frac{2-0.05x}{0.1},\text{ (2)} \\ x\le8,\text{ (3)} \\ y>=22,\text{ (4)} \end{gathered}[/tex]
