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Linear Inequalities in Two Variables

Graph inequalities like 2x + 3 < y on the coordinate plane

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Linear Inequalities

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Complete the chart.
Word Definition
________________ the part on the graph where the shaded areas of the inequalities overlap
Feasible Region _________________________________________________________
________________ a restriction or condition presented in a real-world problem
Linear Programming _________________________________________________________
Vertex Theorem for Regions _________________________________________________________

Linear Inequalities

The first step of graphing linear inequalities is to write the inequality in slope-intercept form.

What is the form that linear inequalities are typically written in? ____________________

What is slope-intercept form? ____________________

Remember: When an inequality is divided or multiplied by a negative number, the sign is reversed.


Complete the following chart.

Inequality Symbol In Words Dashed or solid?
\begin{align*}\textless\end{align*} less than solid
\begin{align*}\textgreater\end{align*} ___________________ ___________________
\begin{align*}\le\end{align*} ___________________ ___________________
\begin{align*}\ge\end{align*} ___________________ ___________________


Once you graph the line, you must decide where to shade. Choose a point not on the line and plug it in. If it is a solution, you shade that area. If not, shade the other area.

Hint: Once graphed, find the y-intercept. If the equation is \begin{align*}\textless\end{align*} or  \begin{align*}\le\end{align*} then shade the region under the y-intercept, if \begin{align*}\textgreater\end{align*} or \begin{align*}\ge\end{align*} shade the region above the y-intercept.


Without graphing, determine if each point is in the shaded region for each inequality.

  1. (–1, 3) and \begin{align*}2x-4y \le -10\end{align*}
  2. (–5, –1) and 
  3. (6, 2) and \begin{align*}2x+3y \ge -2\end{align*}

Determine the inequality that is modeled by each of the following graphs.


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Graphs of Systems of Linear Inequalities

To graph systems of linear inequalities, graph each inequality separately and find where their shaded regions intersect. This region of intersection is called the ______________________.


For each of the following systems of inequalties, a) solve by graphing, and b) determine three points that satisfy the system.

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Applications of Systems of Inequalities

A system of linear inequalities is often used to determine the maximum or minimum values of a situation with multiple constraints. 

In order to solve this type of problem using linear inequalities, follow these steps:

  1. Make a table to organize the given information.
  2. List the constraints of the situation. Write an inequality for each constraint.
  3. Write an equation for the quantity you are trying to maximize (like profit) or minimize (like cost).
  4. Graph the constraints as a system of inequalities.
  5. Find the exact coordinates for each vertex from the graph or algebraically.
  6. Use the Vertex Theorem. Test all vertices of the feasible region in the equation and see which point is the maximum or minimum.


For each graphed region and corresponding equation, find a point at which ‘ \begin{align*}z\end{align*} ’ has a maximum value and a point at which ‘ \begin{align*}z\end{align*} ’ has a minimum value.

  1. \begin{align*}\boxed{z=10x+20y}\end{align*}

    1. \begin{align*}\boxed{z=20x-15y+4}\end{align*}

Beth is knitting mittens and gloves. Each pair must be processed on three machines. Each pair of mittens requires 2 hours on Machine A, 2 hours on Machine B and 4 hours on Machine C. Each pair of gloves requires 4 hours on Machine A, 2 hours on Machine B and 1 hour on Machine C. Machine A, B, and C are available 32, 18 and 24 minutes each day respectively. The profit on a pair of mittens is $8.00 and on a pair of gloves is $10.00. How many pairs of each should be made each day to maximize the profit?

  1. List the constraints and state the profit equation.
  2. Create a graph and identify the feasible region.
  3. Determine what the Beth must do to maximize her profit.


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