Solving Linear Inequalities in Two Variables Adapted from Walch Education

Key Concepts: Inequalities have infinitely many solutions and all the solutions need to be represented. This will be done through the use of shading. A linear inequality in two variables has a half plane as the set of solutions. A half plane is a region containing all points that has one boundary, which is a straight line that continues in both directions infinitely.

Key Concepts continued Sometimes the line or the boundary is part of the solution; this means it’s inclusive. Inequalities that have “greater than or equal to” (≥) or “less than or equal to” (≤) symbols are inclusive. –Use a solid line when graphing the solution to inclusive inequalities. Other times the line or boundary is NOT part of the solution; in other words, it’s non-inclusive. Inequalities that have “greater than” (>) or “less than” (<) symbols are non- inclusive. –Use a dashed line when graphing the solution to non-inclusive inequalities.

Key Concepts continued

Graphing a Linear Inequality in Two Variables Determine the symbolic representation (write the inequality using symbols) of the scenario if given a context. Graph the inequality as a linear equation. If the inequality is inclusive (≤ or ≥), use a solid line. If the inequality is non-inclusive ( ), use a dashed line. Pick a test point above or below the line. If the test point makes the inequality true, shade the half plane that contains the test point. If the test point makes the inequality false, shade the half plane that does NOT contain the test point.

Quick Graphs Using Intercepts Standard Form of Linear Equations and Inequalities Linear equations can also be written as ax + by = c, where a, b, and c are real numbers. An intercept is the point at which the line intersects (or intercepts) the x- or y-axis.

Finding Intercepts The y-intercept is the point at which the line intersects the y- axis. The general coordinates for the y-intercept are (0, y). To solve for the y-intercept in an equation, set x = 0 and solve for y. The x-intercept is the point at which the line intersects the x- axis. The general coordinates for the x-intercept are (x, 0). To solve for the x-intercept in an equation, set y = 0 and solve for x.

Practice # 1 Graph the solutions to the following inequality. y > x + 3 First, graph the inequality as a linear equation. (Since the inequality is non-inclusive, use a dashed line) y = x + 3 The y-intercept is 3 and the slope is 1

We need to shade the appropriate half plane

Choose (0, 0) because this point is easy to substitute into the inequality y > x + 3 (0) > (0) > 3 This is false!

Thanks for Watching !!!!! ~Dr. Dambreville