Graphing Linear Inequalities in Two Variables

Graphing Linear Inequalities The graph of a linear inequality in two variables is the graph of all solutions of the inequality. The boundary line of the inequality divides the coordinate plane into two half-planes: a shaded region which contains the points that are solutions of the inequality, and an unshaded region which contains the points that are not. GRAPHING A LINEAR INEQUALITY The graph of a linear inequality in two variables is a half-plane. To graph a linear inequality, follow these steps. 1 STEP Graph the boundary line of the inequality. Use a dashed line for < or > and a solid line for £ or ³. 2 STEP To decide which side of the boundary line to shade, test a point not on the boundary line to see whether it is a solution of the inequality. Then shade the appropriate half-plane.

*Notice the line is dashed Graphing an Inequality in Two Variables Graph x < 2 Step 1: Start by graphing the line x = 2 Now what points would give you less than 2? Since it has to be x < 2 we shade everything to the left of the line. *Notice the line is dashed

Graphing a Linear Inequality Sketch a graph of y  3 *Notice the line is solid

Some Helpful Hints If the sign is > or < the line is dashed If the sign is  or  the line will be solid When dealing with just x and y. If the sign > or  the shading either goes up or to the right If the sign is < or  the shading either goes down or to the left

Graphing Linear Inequalities in One Variable Graph a) y < –2 and b) x £ 1 in a coordinate plane. SOLUTION Graph the boundary line x = 1. Use a solid line because x £ 1. Graph the boundary line y = –2. Use a dashed line because y < – 2. Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the half-plane to the left of the line. Test the point (0, 0). Because (0, 0) is not a solution of the inequality, shade the half-plane below the line.

Step 1: Put into slope intercept form Using What We Know Sketch a graph of x + y < 3 Step 1: Put into slope intercept form y <-x + 3 Step 2: Graph the line y = -x + 3

Less than means to the left or below. To check it, pick any point that is not on the line. (0,0) is an easy point to use. x + y < 3 Substitute 0 for x and y. 0 + 0 < 3 0 < 3 Decide if this is true or false. Is 0 less than 3? If it is true, you shade on the same side of the line of the point you picked. If it is false, you shade on the opposite side of the line where the point you picked lies.

Graphing Linear Inequalities in Two Variables Graph a) y < 2x and b) 2x – 5y ³ 10. Graph the boundary line 2x – 5y = 10. (Put in slope-intercept form and graph) Use a solid line because 2x – 5y ³ 10. SOLUTION Graph the boundary line y = 2x. Use a dashed line because y < 2x. Test the point (0, 0). Test the point (1, 1). Because (0, 0) is not a solution of the inequality, shade the half-plane below the line. Because (1, 1) is a solution of the inequality, shade the half-plane below the line.

Graphing Inequalities in Two - Variables