Mathematics
Grade 12
15 min
Solve systems of linear and absolute value inequalities by graphing
Solve systems of linear and absolute value inequalities by graphing
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1
Introduction & Learning Objectives
Learning Objectives
Graph a linear inequality in two variables, correctly identifying the boundary line and shaded region.
Graph an absolute value inequality in two variables, correctly identifying the vertex, boundary 'V' shape, and shaded region.
Distinguish between solid and dashed boundary lines based on the inequality symbol.
Apply the test point method to accurately determine the correct region to shade for any inequality.
Graph a system containing both linear and absolute value inequalities.
Identify the solution set of a system of inequalities as the overlapping (doubly-shaded) region on the graph.
Determine if a given ordered pair is a solution to the system by analyzing the final graph.
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Key Concepts & Vocabulary
TermDefinitionExample
System of InequalitiesA collection of two or more inequalities involving the same set of variables. A solution to the system is an ordered pair (x, y) that makes all inequalities in the set true.The set { y < 2x + 1, y ≥ |x - 3| } is a system of one linear and one absolute value inequality.
Boundary Line/CurveThe graph of the equation corresponding to the inequality (e.g., y = mx + b or y = a|x-h|+k). It divides the coordinate plane into two or more regions.For the inequality y > -3x + 4, the boundary line is the line y = -3x + 4.
Solution Region (Feasible Region)The area on the coordinate plane representing all the points (x, y) that are solutions to the system. It is the region where the shading for all individual inequalities overlaps.If one inequality shade...
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Core Formulas
Boundary Line Style Rule
For inequalities with > or <, use a dashed line. For inequalities with ≥ or ≤, use a solid line.
This rule determines whether the points on the boundary line itself are included in the solution set. A dashed line signifies exclusion, while a solid line signifies inclusion.
Shading Rule for y-Inequalities
For y > f(x) or y ≥ f(x), shade ABOVE the boundary. For y < f(x) or y ≤ f(x), shade BELOW the boundary.
This is a quick method for inequalities solved for 'y'. 'Greater than' corresponds to higher y-values (above), and 'less than' corresponds to lower y-values (below). Always verify with a test point if unsure.
Absolute Value Vertex Form
y = a|x - h| + k
Use this form to quickly identify the key featu...
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Challenging
What are the coordinates of all the vertices of the bounded feasible region for the system from Worked Example 2: y ≥ |x - 2| and y < -|x| + 5?
A.(-1.5, 3.5) and (3.5, 1.5)
B.(2, 0) and (0, 5)
C.(2, 0), (0, 5), (-1.5, 3.5), and (3.5, 1.5)
D.The region is unbounded and has no vertices.
Challenging
Consider the system {y > a|x|, y < 4}. How does the area of the triangular feasible region change as the positive value of 'a' increases?
A.The area increases.
B.The area decreases.
C.The area remains constant.
D.The area change depends on the y-intercept.
Challenging
A student is graphing the system {y ≤ -|x| + 4, y > x - 3}. They test the point (3, 0) and find it satisfies both inequalities, concluding it is in the solution set. What is the flaw in their reasoning?
A.The point (3, 0) is on the boundary of y > x - 3, so it cannot be a solution.
B.The test of the first inequality is incorrect: 0 ≤ -|3| + 4 is false.
C.The test of the second inequality is incorrect: 0 > 3 - 3 is false.
D.The student made no error; (3, 0) is a solution.
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