CHAPTER TWO: LINEAR EQUATIONS AND FUNCTIONS ALGEBRA TWO Section Linear Inequalities in Two Variables

LEARNING GOALS Goal One - Graph linear inequalities in two variables. Goal Two - Use linear inequalities to solve real-life problems.

VOCABULARY linear inequality Ax + By C Ax + By > C A linear inequality in two variables can be written in one of the following forms: Ax + By C, and Ax + By > C. solution An ordered pair (x, y) is a solution of a linear inequality if the inequality is true when the values for x and y are substituted into the inequality.

VOCABULARY graph A graph of a linear inequality in two variables is the graph of all solutions of the inequality. half-planes The boundary line of the inequality divides the coordinate plane into two half-planes; a shaded region containing the points that are solutions of the inequality, and an unshaded region that contains the points that are not.

PROBLEM: Check whether the point (0, -6) is a solution of -x + 2y < 6 SOLUTION -x + 2y < 6 Write the original inequality. Checking Solutions of Inequalities The point (0, -6) IS a solution is this inequality. -(0) + 2(-6) < 6 Substitute 0 for x and -6 for y. -12 < 6 Simplify. TRUE STATEMENT!

PROBLEM: Check whether the point (2, 4) is a solution of -x + 2y < 6 SOLUTION -x + 2y < 6 Write the original inequality. Checking Solutions of Inequalities The point (2, 4) is NOT a solution is this inequality. -(2) + 2(4) < 6 Substitute 2 for x and 4 for y. 6 < 6 Simplify. FALSE STATEMENT!

PROBLEM: Graph 2y > 6 and x > -5 in a coordinate plane. Graphing Linear Inequalities in One Variable 2y > 6 y > 3 Graph the boundary line y = 3. Use a dashed line because y > 3 (if it had been > then it would be a solid line)

PROBLEM: Graph 2y > 6 and x > -5 in a coordinate plane. Graphing Linear Inequalities in One Variable 2y > 6 y > 3 Shade above the dashed line because y > 3.

PROBLEM: Graph 2y > 6 and x > -5 in a coordinate plane. Graphing Linear Inequalities in One Variable 2y > 6 y > 3 x > -5 Graph the boundary line x = -5. Use a solid line because x > -5

PROBLEM: Graph 2y > 6 and x > -5 in a coordinate plane. Graphing Linear Inequalities in One Variable 2y > 6 y > 3 x > -5 Shade to the right of the line because x > -5.

PROBLEM: Graph 2y > 6 and x > -5 in a coordinate plane. Graphing Linear Inequalities in One Variable 2y > 6 y > 3 x > -5 The purple area represents those points that are common to both inequalities.

PROBLEM: Graph 2y > 6 and x > -5 in a coordinate plane. Graphing Linear Inequalities in One Variable 2y > 6 y > 3 x > -5 (0, 0) TEST THE FOLLOWING This point IS a solution for x > -5 but NOT for y > 3.

PROBLEM: Graph 2y > 6 and x > -5 in a coordinate plane. Graphing Linear Inequalities in One Variable 2y > 6 y > 3 x > -5 (3, 5) TEST THE FOLLOWING This point IS a solution for both x > -5 and y > 3.

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: STEP 1:Graph the boundary line of the inequality. Use a dashed line for and a solid line for. STEP 1: Graph the boundary line of the inequality. Use a dashed line for and a solid line for. STEP 2: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. STEP 2: 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.

PROBLEM: Graph y > -2x - 5 Graphing Linear Inequalities in Two Variables Plot the y-intercept b = -5 Find the next point by using slope 2/-1, move two units up and one unit to the left.

PROBLEM: Graph y > -2x - 5 Graphing Linear Inequalities in Two Variables Draw a line connecting the two points. Make it dashed because y > -2x - 5

PROBLEM: Graph y > -2x - 5 Graphing Linear Inequalities in Two Variables TEST THE POINT (0, 0) TO DETERMINE IF IT IS A SOLUTION. y > -2x - 5 (0) > -2(0) - 5 (0) > - 5 This is a TRUE statement, so (0, 0) IS a solution. Shade the area where you find (0, 0)

PROBLEM: Graph y > -2x - 5 Graphing Linear Inequalities in Two Variables TEST THE POINT (0, 0) TO DETERMINE IF IT IS A SOLUTION. y > -2x - 5 (0) > -2(0) - 5 (0) > - 5 This is a TRUE statement, so (0, 0) IS a solution. Shade the area where you find (0, 0)

ASSIGNMENT pg #15, #19, #21, #27, #31, #57, #61, & #63. pg #15, #19, #21, #27, #31, #57, #61, & #63.