Systems of Linear Inequalities with Three Inequalities AII, 2.0: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.

Systems of Linear Inequalities Objective Key Words Graph, write, and use a system of linear inequalities. Three Inequalities Why we need this? It will prepare you for college in related fields such as math and science. System of linear inequalities in two variables Think about what a system of equations looks like Solution of a system of linear inequalities Think about shaded regions Graph of a system of linear inequalities Think about dashed lines, solid lines, and shaded regions

Pre-requisite Check Graph system of inequalities y<-4x x+3y≥3 Check Solutions of Inequalities y<-4x x+3y≥3 Check whether (4,2) is a solution of the system of inequalities. 𝑥+𝑦>−2 𝑥−𝑦≥4 𝑥−3𝑦≤4 𝑥>1

You-Try Exercises From Yesterday! Graph and test a solution point (ie. (0,0)) ≤ y x 2 + ANSWER

You-Try Exercises From Yesterday! Check whether (3, 1) is a solution of the system of inequalities. – < y 2 1 > x + 1. ANSWER solution 2. 4 y – 2x + ≤ 6 2y > x ANSWER not a solution 3. 4 y x – ≥ 8 4y > 3x + ANSWER not a solution

Graphing a System of Linear Inequalities Step 1: Graph the boundary lines of the inequalities. Use a dashed line for an inequality with < or >. Use a solid line for an inequality with ≤ or ≥. Step 2: Shade the half-planes for the inequalities. The graph of the system is the region common to all the half-planes. 𝑦≥−1 𝑥<2 𝑦>𝑥

YOU DO: Graph the system. Checkpoint Graph a System of Two Inequalities From Yesterday! YOU DO: Graph the system. 3. + y x 4 > y 2x 3 ≥ – ANSWER

The inequality implies that the region is on and above the line ≥ y 2 Example 1 Graph a System of Three Inequalities I DO: Graph the system. ≥ x 2 Inequality 1 ≥ y 2 – Inequality 2 ≤ 3x 2y 12 + Inequality 3 SOLUTION The inequality implies that the region is on and to the right of the line ≥ x 2 = . The inequality implies that the region is on and above the line ≥ y 2 – = .

The inequality implies that the region is on and below the line . ≤ Example 1 Graph a System of Three Inequalities The inequality implies that the region is on and below the line . ≤ 3x 2y 12 + = 3x 2y 12 + The graph of the system is the shaded triangular region shown at the bottom. 9

I DO #4 and YOU DO #5: Graph the system. Checkpoint Graph a System of Three Inequalities I DO #4 and YOU DO #5: Graph the system. 4. ≥ x y x y 5 + < ANSWER 5. ≥ x y 2 2x + < 6 ANSWER

YOU DO: Graph the system. Checkpoint Graph a System of Three Inequalities YOU DO: Graph the system. 6. ≤ y 5 < x 2 3x + ≥ ANSWER

Conclusion Summary Assignment How do you find the solution of a system of linear inequalities? Graph each inequality. The solution is the overlapping shaded region, if any. To be completed in class Pg 188 #(20,26,42,54)