6.5 Graphing Linear Inequalities in Two Variables Wow, graphing really is fun!

Learning Goal #1 for Focus 4 (HS.A-CED.A.2, HS.REI.ID.10 & 12, HS.F-IF.B.6, HS.F-IF.C.7, HS.F-LE.A.2): The student will understand that linear relationships can be described using multiple representations. 4 3 2 1 In addition to level 3.0 and above and beyond what was taught in class,  the student may: · Make connection with other concepts in math · Make connection with other content areas. The student will understand that linear relationships can be described using multiple representations. - Represent and solve equations and inequalities graphically. - Write equations in slope-intercept form, point-slope form, and standard form. - Graph linear equations and inequalities in two variables. - Find x- and y-intercepts. The student will be able to: - Calculate slope. - Determine if a point is a solution to an equation. - Graph an equation using a table and slope-intercept form.   With help from the teacher, the student has partial success with calculating slope, writing an equation in slope-intercept form, and graphing an equation. Even with help, the student has no success understanding the concept of a linear relationships.

What is a linear inequality? A linear inequality in x and y is an inequality that can be written in one of the following forms. ax + by < c ax + by ≤ c ax + by > c ax + by ≥ c

An ordered pair (a, b) is a solution of a linear equation in x and y if the inequality is TRUE when a and b are substituted for x and y, respectively. For example: is (1, 3) a solution of 4x – y < 2? 4(1) – 3 < 2 1 < 2 This is a true statement so (1, 3) is a solution.

Check whether the ordered pairs are solutions of 2x - 3y ≥ -2. a Check whether the ordered pairs are solutions of 2x - 3y ≥ -2. a. (0, 0) b. (0, 1) c. (2, -1) (x, y) Substitute Conclusion A (0,0) 2(0) – 3(0) = 0 ≥ -2 (0,0) is a solution. B (0,1) 2(0) – 3(1) -3 ≥-2 (0, 1) is NOT a solution. C (2,-1) 2(2) – 3(-1) 7 ≥ -2 (2, -1) is a solution.

Graph the inequality 2x – 3y ≥ -2 Every point in the shaded region is a solution of the inequality and every other point is not a solution. 3 2 1 -3 -2 -1 1 2 3 4 -1 -2 -3

Steps to graphing a linear inequality: Sketch the graph of the corresponding linear equation. Use a dashed line for inequalities with < or >. Use a solid line for inequalities with ≤ or ≥. This separates the coordinate plane into two half planes.

Test a point in one of the half planes to find whether it is a solution of the inequality. If the test point is a solution, shade its half plane. If not shade the other half plane.

Sketch the graph of 6x + 5y ≥ 30 Use x- and y-intercepts: (0, 6) & (5, 0) This will be a solid line. Test a point. (0,0) 6(0) + 5(0) ≥ 30 0 ≥ 30 Not a solution. Shade the side that doesn’t include (0,0). 6 4 2 -6 -4 -2 2 4 6 8 -2 -4 -6

Sketch the graph y < 6. This will be a dashed line at y = 6. Test a point. (0,0) 0 < 6 This is a solution. Shade the side that includes (0,0). 6 4 2 -6 -4 -2 2 4 6 8 -2 -4 -6

Sketch the graph of 2x – y ≥ 1 Use x- and y-intercepts: (0, -1) & (1/2, 0) This will be a solid line. Test a point. (0,0) 2(0) - 0 ≥ 1 0 ≥ 1 Not a solution. Shade the side that doesn’t include (0,0). 3 2 1 -3 -2 -1 1 2 3 4 -1 -2 -3