3.3 Solving Systems of Inequalities by Graphing

Objectives:1. Solve systems of inequalities by

graphing2.Determine the coordinates of the

vertices of a region formed by the graph of a system of inequalities

System of Inequalities• The solution is the set of ordered pairs that satisfy

all of the inequalities in the system• Because inequalities are involved, there is an

infinite number of solutions. The best way to represent this many solutions is graphically.

• Graph each line and shade according to the inequality.

Helpful hints:≥ shade above the line < ≤ shade below the line> or < dotted line ≥ ≤ solid line

Example

• Solve each system of inequalities by graphing.

• y≥2x-3• y<-x+2(already solved for y!)

The blue region represents the solution set.

Find the coordinates of the vertices of the figure formed by each system of inequalities.

• 2x-y≥-1• x+y≤4• x+4y≥4Solve for y-y≥-2x-1 y≤2x+1y≤-x+4 y≤-x+44y≥-x+4 y≥-¼x+1

Continued• y≤2x+1• y≤-x+4• y≥-¼x+1Find the solution for each pair of

equations.Substitution is easy since they are

already solved for y.2x+1=-x+4 3x=3 x=1y=2(1)+1y=3 These lines cross at (1,3)

• y≤-x+4• y≥-¼x+1-x+4= -¼x+1 These 3=¾x lines 4=x cross y=-4+4 at y=0 (4,0)y≤2x+1y≥-¼x+12x+1=-¼x+12¼x=0 x=0 y=2(0)+1These lines cross at (0,1)

Another Example

• y>-2• x+y≤3

Solve 2nd equation for yy>-2y≤-x+3

The solution in the blue region

Absolute Value• Solve by graphing.• y≤-x+1• |x+1|<3Remember that an

absolute value equation has 2 possible solutions.

Don’t forget to flip inequality and change sign on 2nd inequality.

x+1<3 and x+1>-3X<2 and x>-4

Homework:Page 126, 13-29 odd