Entry Task Solve for y 1) 2x + -3y < 12 2) x > ½ y - 7

3.3 SYSTEMS OF INEQUALITIES Learning Target: I can solve Linear Systems of Inequalities by Graphing Success Criteria: I will be able to graph 2 inequalities and find the “overlap”

Graphing Method Example: Graph the inequalities on the same plane: x + y < 6 and 2x - y > 4. Before we graph them simultaneously, let’s look at them separately. Graph of x + y < 6. --->

So what happens when we graph both inequalities simultaneously? Graphing Method This is: 2x - y > 4. So what happens when we graph both inequalities simultaneously?

Coolness Discovered! Wow! The solution to the system is the brown region - where the two shaded areas coincide. The green region and red regions are outside the solution set.

Solving Systems of Linear Inequalities We show the solution to a system of linear inequalities by graphing them. This process is easier (but not required) if we put the inequalities into Slope-Intercept Form, y = mx + b. Why? If y < x then shade below the line If y > x then shade above the line

Solving Systems of Linear Inequalities Graph the line using the y-intercept & slope. If the inequality is < or >, make the lines dotted. If the inequality is < or >, make the lines solid.

Solving Systems of Linear Inequalities The solution also includes points not on the line, so you need to shade the region of the graph: Use a test point (0,0) if available. What makes (0,0) unavailable?

Solving Systems of Linear Inequalities Example: a: 3x + 4y > - 4 b: x + 2y < 2 Put in Slope-Intercept Form:

Solving Systems of Linear Inequalities Example, continued: Graph each line, make dotted or solid and shade the correct area. a: dotted shade above b: dotted shade below

Solving Systems of Linear Inequalities a: 3x + 4y > - 4

Solving Systems of Linear Inequalities a: 3x + 4y > - 4 b: x + 2y < 2

Solving Systems of Linear Inequalities a: 3x + 4y > - 4 b: x + 2y < 2 The area between the green arrows is the region of overlap and thus the solution.

Things to remember….. There is one special case. Sometimes you may have to reverse the direction of the inequality sign!! That only happens when you multiply or divide both sides of the inequality by a negative number.

Example Graph the following linear system of inequalities. Use the slope and y-intercept to plot two points for the first inequality. x y Draw in the line. For ≥ use a solid line. Pick a point and test it in the inequality. Shade the appropriate region.

Example Graph the following linear system of inequalities. y The region above the line should be shaded. Now do the same for the second inequality.

Example Graph the following linear system of inequalities. Use the slope and y-intercept to plot two points for the second inequality. x y Draw in the line. For < use a dashed line. Pick a point and test it in the inequality. Shade the appropriate region.

Example Graph the following linear system of inequalities. y The region below the line should be shaded.

Example Graph the following linear system of inequalities. The solution to this system of inequalities is the region where the solutions to each inequality overlap. This is the region above or to the left of the green line and below or to the left of the blue line. Shade in that region. x y

You Try It Graph the following linear systems of inequalities.

Problem 1 x y Use the slope and y-intercept to plot two points for the first inequality. Draw in the line. Shade in the appropriate region.

Problem 1 x y Use the slope and y-intercept to plot two points for the second inequality. Draw in the line. Shade in the appropriate region.

Problem 1 x y The final solution is the region where the two shaded areas overlap (purple region).

Assignment pg 153-154 (10pts) Homework: P.153 #11,13,17,19,20,21,32, Challenge: # 56 – You may see something like this on the test