Solve systems of linear and absolute value inequalities by graphing

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. How can a GPS define a search area for a lost signal that is '...

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...

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...