Systems of Equations and Inequalities

Suppose that you are the manager of a sheet metal shop. A customer asks you to manufacture $10,000$ boxes, each box being open on top. The boxes are required to have a square base and a $9$-cubic-foot capacity. You construct the boxes by cutting out a square from each corner of a square piece of sheet metal and folding along the edges.

Part (a): What are the dimensions of the square to be cut if the area of the square piece of sheet metal is $100$ square feet?

Part (b): Could you make the box using a smaller piece of sheet metal? Make a list of the dimensions of the box for various pieces of sheet metal.

 Solve the inequality $3 x+4<8-x$

Graph the equation $3x-2 y=6$

Graph the equation: $x^2+y^2=9$

Graph the equation: $y=x^2+4$

Graph the equation: $x^2+y^2=9$

Graph each inequality by hand.

$y\le 2$

Graph each inequality by hand. 

$2x+y\ge 6$

Graph each inequality by hand. 

$3x+2y\le 6$

Graph each inequality by hand. 

$x^2+y^2>1$

Graph each inequality by hand. 

$x^2+y^2\le 9$

Graph each inequality by hand. 

$y\le x^2-1$

Graph each inequality by hand. 

$y>x^2+2$

Graph each inequality by hand. 

$xy\ge 4$

Graph each inequality by hand. 

$xy\le 1$

Graph each system of linear inequalities by hand. Verify your results using a graphing utility.

$$\begin{gathered} x+y\le 2 \\ 2x+y\ge 4 \end{gathered}$$

Graph each system of linear inequalities by hand. Verify your results using a graphing utility. 

$$\begin{gathered} 3x-y\ge 6 \\ x+2y\le 2 \end{gathered}$$

Graph each system of linear inequalities by hand. Verify your results using a graphing utility. 

$$\begin{gathered} 2x-y\le 4 \\ 3x+2y\ge -6 \end{gathered}$$ 

Graph each system of linear inequalities by hand. Verify your results using a graphing utility.  

$$\begin{gathered} 4x-5y\le 0 \\ 2x-y\ge 2 \end{gathered}$$

Graph each system of linear inequalities by hand. Verify your results using a graphing utility.   

$$\begin{gathered} 2x-3y\le 0 \\ 3x+2y\le 6 \end{gathered}$$

Graph each system of linear inequalities by hand. Verify your results using a graphing utility.    

$$\begin{gathered} 4x-y\ge 2 \\ x+2y\ge 2 \end{gathered}$$

In Problems 23–34, graph each system of linear inequalities by hand. Verify your results using a graphing utility 

$\left\{\begin{array}{r}x-2y\le 6\\ 2x-4y\ge 0\end{array}\right.$

In Problems 23–34, graph each system of linear inequalities by hand. Verify your results using a graphing utility  

$\left\{\begin{array}{l}x+4y\le 8\\ x+4y\ge 4\end{array}\right.$

In Problems 23–34, graph each system of linear inequalities by hand. Verify your results using a graphing utility 

$\left\{\begin{array}{l}2x+y\ge -2\\ 2x+y\ge 2\end{array}\right.$

In Problems 23–34, graph each system of linear inequalities by hand. Verify your results using a graphing utility

$\left\{\begin{array}{l}x-4y\le 4\\ x-4y\ge 0\end{array}\right.$

In Problems 23–34, graph each system of linear inequalities by hand. Verify your results using a graphing utility:

$\left\{\begin{array}{l}2x+3y\ge 6\\ 2x+3y\le 0\end{array}\right.$

In Problems 23–34, graph each system of linear inequalities by hand. Verify your results using a graphing utility

$\left\{\begin{array}{l}2x+y\ge 0\\ 2x+y\ge 2\end{array}\right.$

In Problems 35–42, graph each system of inequalities by hand. 

$\left\{\begin{array}{l}{x}^2+y^2\le 9\\ x+y\ge 3\end{array}\right.$

In Problems 35–42, graph each system of inequalities by hand. 

$\left\{\begin{array}{l}{x}^2+y^2\ge 9\\ x+y\le 3\end{array}\right.$

In Problems 35–42, graph each system of inequalities by hand 

$\left\{\begin{array}{l}y\ge x^2-4\\ y\le x-2\end{array}\right.$

In Problems 35–42, graph each system of inequalities by hand 

$\left\{\begin{array}{l}{y}^2\le x\\ y\ge x\end{array}\right.$

In Problems 35–42, graph each system of inequalities by hand 

$\left\{\begin{array}{l}{x}^2 + y^2 \le 16 \\ y\ge x^2 - 4\end{array}\right.$

In Problems 35–42, graph each system of inequalities by hand 

$\left\{\begin{array}{l}{x}^2 + y^2 \le 25\\ y\ge x^2 - 5\end{array}\right.$

In Problems 35–42, graph each system of inequalities by hand 

$\left\{\begin{array}{l}xy\ge 4\\ y\ge x^2+1\end{array}\right.$

In Problems 35–42, graph each system of inequalities by hand 

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

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points 

$\left\{\begin{array}{l}x\ge 0 \\ y\ge 0\\ 2x+y\le 6\\ x+2y\le 6\end{array}\right.$

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points  

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 4\\ 2x+3y\ge 6\end{array}\right.$

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points  

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ 2x+y\ge 4\end{array}\right.$

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points. 

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 3x+y\le 6\\ 2x+y\le 2\end{array}\right.$

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points. 

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 2\\ 2x+3y\le 12\\ 3x+y\le 12\end{array}\right.$

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points.

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\ge 1\\ x+y\le 7\\ 2x+y\le 10\end{array}\right.$

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points

$49.\left\{\begin{array}{l} x\ge 0\\ y\ge 0\\ x+y\ge 2\\ x+y\le 8\\ 2x+y\le 10\end{array}\right.$

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points

$50.\left\{\begin{array}{l} x\ge 0\\ y\ge 0\\ x+y\ge 2\\ x+y\le 8\\ x+2y\ge 1\end{array}\right.$

In Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points

$51.\left\{\begin{array}{l} x\ge 0\\ y\ge 0\\ x+2y\ge 1\\ x+2y\le 10\end{array}\right.$

Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points

 $52.\left\{\begin{array}{l} x\ge 0\\ y\ge 0\\ x+2y\ge 1\\ x+2y\le 10\\ x+y\ge 2\\ x+y\le 8\end{array}\right.$

In Problems 53–56, write a system of linear inequalities for the given graph.

In Problems 53–56, write a system of linear inequalities for the given graph.

54.

In Problems 53–56, write a system of linear inequalities for the given graph.

In Problems 53–56, write a system of linear inequalities for the given graph.

Financial Planning A retired couple has up to \(50,000 to

invest. As their financial adviser, you recommend that they

place at least \)35,000 in Treasury bills yielding 1% and at

most $10,000 in corporate bonds yielding 3%.

(a) Using x to denote the amount of money invested in

Treasury bills and y the amount invested in corporate

bonds, write a system of linear inequalities that describes

the possible amounts of each investment.

(b) Graph the system and label the corner points.