GR-5 Linear Inequalities

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Section GR-5 Linear Inequalities

Worksheet Linear Inequalities and Systems

For this outcome Specifically, for this outcome, students should be able to:

Solve linear inequalities in one variable and graph their solutions.
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Solve systems of linear inequalities in two variables and graph their solutions.
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Model real-world problems using linear inequalities and systems of linear inequalities.
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Identify the feasible region in a system of linear inequalities and determine whether a given point is a solution to the system.
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Linear Inequalities

A linear inequality in one variable is an inequality that can be written in the form \(ax + b \lt c\text{,}\) where \(a\text{,}\) \(b\text{,}\) and \(c\) are real numbers and \(a \neq 0\text{.}\) The solution to a linear inequality is a set of values that satisfy the inequality. To solve a linear inequality, isolate the variable on one side of the inequality sign. When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the inequality sign.

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Systems of Linear Inequalities

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A system of linear inequalities in two variables is a set of two or more linear inequalities in two variables that are solved simultaneously. The solution to a system of linear inequalities is the set of all points that satisfy all inequalities in the system. The solution set is often represented graphically as a region in the coordinate plane and the corner points of that region are the vertices of the feasible region.

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To solve a system of linear inequalities, graph each inequality on the same coordinate plane. The solution is the intersection of the regions defined by each inequality. The vertices of the feasible region are found by solving systems of equations formed by pairs of boundary lines.

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Worksheet GR-5 Linear Inequalities

Solving Systems of Linear Inequalities

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Section GR-5 Rubric

Worksheet Worksheet

Before attempting the GR-5 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.

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Table 18. GR5 Rubric

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Criteria	M	P	R	N
Simplifying Algebraic Expressions	All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner	No algebraic errors OR some expressions are not simplified.	There are some algebraic errors and some expressions were not simplified.	There are some algebraic errors and some expressions were not simplified.
Basics of Graphing	Graph is labeled with units for each axis. The graph includes all of the important features (intercepts, asymptotes, deleted points, etc). The graph is appropriately drawn to accommodate all features.	Graph is labeled with units for each axis and includes most of the important features or the graph has all of the important features but is not labeled.	Graph is not labeled and has some of the important features.	No graph was given or the given graph is not labeled and does not have the correct features of the function.
Problem Structure and Mathematical Communication	Problems have a clear beginning, middle, and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and notation and provides a clear solution.	Problems have clear beginnings and ends. Work progresses from one step to the next. The work provided uses algebraic methods, correct notation, and provides a solution.	Parts of the mathematical structure are missing, or steps in the progress of the solution are missing. There are errors in the mathematical notation. The work provided uses algebraic methods and provides a solution.	There is not a clear structure to the solution, notation is used incorrectly, and it is unclear what methods are being used.
Graph a Linear Inequality	Provided the correct line, correct density (solid or dashed), correct intercepts are show and the correct region is indicated.	Provided the correct line, correct intercepts are show and the correct region is indicated.	Provided a line with a region is indicated.	No graph given.
Graph a System of Inequalities	For each inequality in the system, a correct line is graphed with correct solid or dotted notation; the correct solution region is indicated, and all corner points are clearly identified.	For each inequality in the system, a line is graphed with solid or dotted notation; a solution region is indicated, and corner points are identified; minor errors in the solution.	For some of the inequalities in the system, a line is graphed; a solution region is indicated; and multiple errors in the solution.	No graph given; missing corner points of the solution.
Solve a Linear Programming Problem	For each inequality in the system, a correct line is graphed with correct solid or dotted notation; the correct solution region is indicated, and all corner points are identified. The objective function is evaluated at the corner points to find the optimal solution.	For each inequality in the system, a line is graphed with solid or dotted notation; a solution region is indicated, and corner points are identified; minor errors in the solution. The objective function is evaluated at the corner points to find a solution.	For some of the inequalities in the system a line is graphed; a solution region is indicated; multiple errors in the solution; errors in finding the optimal solution.	No system, objective function or graph were given; missing corner points of the solution; no optimal solution is found.

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Worksheet GR-5 Graphing Linear Inequalities

This outcome covers graphing linear inequalities. You should be able to graph linear inequalities on a coordinate plane and identify the solution region.

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For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:

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Objectives

Simplifying Expressions: Simplify algebraic expressions by combining like terms and applying the distributive property. Answers should be completely simplified.
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Basics of Graphing: Give a graph with appropriate scales and labels. The line should have the correct intercepts and general behavior.
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Problem structure: Solution should be organized and easy to follow, with clear, complete steps that follow progressively from one to the next.
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Mathematical Communication: Solutions should be presented in a clear and logical manner, with appropriate use of mathematical notation and terminology. Equations should have at most one equal sign and expressions on both sides.
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Graph a linear inequality on a coordinate plane.
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Identify the solution region for a linear inequality.
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Graph a system of linear inequalities, identify the solution region and label the corner points.
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1.

Solve the inequality:: \(4x-2y-3\le 5\text{.}\)

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Hint.

Determine if the boundary line is inclusive or non-exclusive. Then graph the line and shade the appropriate region.

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Answer.

Diagram Exploration Keyboard Controls

Key	Action
Enter, A	Activate keyboard driven exploration
B	Activate menu driven exploration
Escape	Leave exploration mode
Cursor down	Explore next lower level
Cursor up	Explore next upper level
Cursor right	Explore next element on level
Cursor left	Explore previous element on level
X	Toggle expert mode
W	Extra details if available
Space	Repeat speech
M	Activate step magnification
Comma	Activate direct magnification
N	Deactivate magnification
Z	Toggle subtitles
C	Cycle contrast settings
T	Monochrome colours
L	Toggle language (if available)
K	Kill current sound
Y	Stop sound output
O	Start and stop sonification
P	Repeat sonification output

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2.

Solve the system of inequalities: \(\begin{cases} 2x+y\le8\\y\le4\\x,y\ge0 \end{cases}\text{.}\)

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Hint.

For each inequality, determine the boundary line and the region that satisfies the inequality. Once you have graphed all inequalities, the solution region will be the intersection of all shaded areas. Find the corner points of the solution region.

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