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Solving Linear Inequalities in Two Variables

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.

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Solving Linear Inequalities in Two Variables

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  1. Solving Linear Inequalities in Two Variables Adapted from Walch Education

  2. 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.

  3. 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.

  4. Key Concepts continued

  5. 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 (< or >), 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.

  6. 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.

  7. 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.

  8. 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

  9. We need to shade the appropriate half plane

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

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

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