+ Solving Linear Inequalities Mr. Smith IM3
+ Inequality Symbols < > Less than Greater than Less than or equal to Greater than or equal to Not equal to
+ Linear Inequality n Inequality with one variable to the first power. for eample: 2-3<8 n A solution is a value of the variable that makes the inequality true. could equal -3, 0, 1, etc.
+ Transformations for Inequalities n Add/subtract the same number on each side of an inequality n Multiply/divide by the same positive number on each side of an inequality n If you multiply or divide by a negative number, you MUST flip the inequality sign!
E: Solve the inequality. 2-3<8 +3 +3 2<11 2 2 11 < 2 3 + 7 13 3 6 2 Flip the sign after dividing by the -3!
+ Graphing Linear Inequalities n Remember: < and > signs will have an open dot o and signs will have a closed dot graph of 11 < 2 graph of 2 4 5 6 7-3 -2-1 0
Eample: Solve and graph the solution. + 7 + 9 10 12 9 3 12 21 3 7 6 7 8 9
+ Compound Inequality n An inequality joined by and or or. Eamples and 3 < 1 or 2 or > 4-4 -3-2 -1 0 1 2 think between -3-2 -1 0 1 2 3 4 5 think oars on a boat
+ Eample: Solve & graph. -9 < t+4 < 10-13 < t < 6 Think between! -13 6
Last eample! Solve & graph. + -6+9 < 3 or -3-8 > 13-6 < -6-3 > 21 > 1 or < -7 Flip signs Think oars -7 1
+ Graphing Linear Inequalities in Two Variables Mr. Smith IM3
Epressions of the type + 2y 8 and 3 y > 6 are called linear inequalities in two variables. A solution of a linear inequality in two variables is an ordered pair (, y) which makes the inequality true. Eample: (1, 3) is a solution to + 2y 8 since (1) + 2(3) = 7 8.
The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions. y Eample: The solution set for + 2y 8 is the shaded region. 2 2 The solution set is a half-plane. It consists of the line + 2y 8 and all the points below and to its left. The line is called the boundary line of the half-plane.
If the inequality is or, the boundary line is solid; its points are solutions. Eample: The boundary line of the solution set of 3 y 2 is solid. 3 y < 2 y 3 y = 2 3 y > 2 If the inequality is < or >, the boundary line is dotted; its points are not solutions. y Eample: The boundary line of the solution set of + y < 2 is dotted.
A test point can be selected to determine which side of the half-plane to shade. Eample: For 2 3y 18 graph the boundary line. The solution set is a half-plane. Use (0, 0) as a test point. (0, 0) is a solution. So all points on the (0, 0) side of the boundary line are also solutions. Shade above and to the left of the line. -2 y (0, 0) 2
To graph the solution set for a linear inequality: 1. Graph the boundary line. 2. Select a test point, not on the boundary line, and determine if it is a solution. 3. Shade a half-plane.
Eample: Graph the solution set for y > 2. 1. Graph the boundary line y = 2 as a dotted line. 2. Select a test point not on the line, say (0, 0). (0) 0 = 0 > 2 is false. (0, 0) y (0, -2) (2, 0) 3. Since this is a not a solution, shade in the half-plane not containing (0, 0).
Solution sets for inequalities with only one variable can be graphed in the same way. Eample: Graph the solution set for < - 2. - 4 y 4 4 Eample: Graph the solution set for 4. - 4-4 y 4 4-4
A solution of a system of linear inequalities is an ordered pair that satisfies all the inequalities. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red still appears, you may have to delete the image and then insert it again. Eample: Find a solution for the system. (5, 4) is a solution of + y > 8. (5, 4) is also a solution of 2 y 7. + y > 8 2 y 7 Since (5, 4) is a solution of both inequalities in the system, it is a solution of the system.
The set of all solutions of a system of linear inequalities is called its solution set. To graph the solution set for a system of linear inequalities in two variables: 1. Shade the half-plane of solutions for each inequality in the system. 2. Shade in the intersection of the half-planes.
Eample: Graph the solution set for the system Graph the solution set for + y > 8. + y > 8 2 y y 7 Graph the solution set for 2 y 7. The intersection of these two half-planes is the wedge-shaped region at the top of the diagram. 2 2
Eample: Graph the solution set for the system of linear inequalities: Graph the two half-planes. 2 3y 12 2 + 3y 6 y -2 + 3y 6 The two half-planes do not intersect; therefore, the solution set is the empty set. 2 2 2 3y 12
(1) (2) (3) (4) Eample: Graph the solution set for the linear system. 2 + 3y 3 6 + y 1 2 y 1 Graph each linear inequality. (1) - 4 The solution set is the intersection of all the half-planes. (3) (2) 4-4 y 4 (4)