Plan Systems Inequalities

Plan: Solving systems of inequalities – 2009 • Objectives By the end of this lesson, students will – write a system of inequalities based on a model situation – be able to identify the solution region of a system of inequalities – test solutions to systems of inequalities • Prep Work – Arrange a Smart Board file with graphs ready to be arranged so that one inequality and another can be overlayed, to suggest the solution. The visual is a better explanation of the idea. • Motivator – Basketball Practice It’s basketball practice. Coach has two rules, first shoot at most 60 shots, second, make more than 100 points. List as many solutions as you can that do not break the coaches rules. – Can you ask the question a different way? – How do you know if you’ve found a solution? x – How many solutions are there? • Agenda – (5 min) Basketball Practice Introduction – (5 min) Reflect: Can we apply a system of equations to this problem? – Model Write and solve by graphing a system of inequalities for the problem – Vocab and Notes – Practice • Model Solving the Basketball Problem with a system of inequalities In order to solve this problem elegantly, we can represent it using a system of inequalities.

x + y ≤ 60 2x + 3y > 100 Graph them individually and notice where solutions to one inequality overlap with solutions to the other. Smart Board suggestions, do two separate graphs, then copy one and move it onto the other 13 12 11 10 9 8 7 6 5 4 3 2 1 0

13 12 11 10 9 8 7 6 5 4 3 2 1 0

• Vocab and Notes When solving inequalities, students should be familiar with the general vocabulary of inequalities, such as – Boundary line: a line on the coordinate plane that separates the plane into two regions – Half Planes one of the two regions on the coordinate plane separated by a boundary line

– Solution Sets The set of all replacements for the variable(s) in an open sentence that result in a true sentence. Students should be familiar with idea of solving systems of inequalities: the intersection of two distinct half planes which may overlap, and yield a solution region. It may be helpful to sketch this as a rule like Identify the solution region of the system of inequalities.

x>0 y≤0 The solution region is quadrant IV

x>0 x < −3 There is no solution region, there are no solutions to this system of inequalities

y<5 y≤3 The solution region is y ≤ 3 • Practice Direct students to work through the activity attached. Solutions at the end. • Review Ask students to sketch systems with no solution, a line as the solution, a small solution region, an unbounded solution region.

Systems of Inequalities – 2009 By the end of this activity, you will be able to solve systems of inequalities, and apply them to real world situations. 1. Make a sketch to represent an example system of equations, label the boundary lines, and solution region.

4. Solve each system of inequalities by shading the solution region, if no solution region exists, say so.

2. The system of inequalities

(a) b b

A

2x + y > 5 y <2−x b

B (b) y ≥ 4x + 2 b

C

x
has an infinite number of solutions. Instead of listing each one, we identify the region on a coordinate plane where all the solutions lie. Shading is a visual strategy for defining all possible solutions to the system. However, we can identify individual solutions very easily from the shading. Take the point (−3, 2) in the shaded region, (−3, 2) is a solution to the system because (2) ≥ 4(−3) + 2 and (−3) < (2) + 2. (a) Show that points A, B, and C are not solutions to the system using substitution 3. With the system of inequalities below y > 2x − 3 y <7−x

y >x−2 y ≤x−4 (c)

x≤3 y≥4 5. Janet is unlucky because her parents love systems of inequalities. Unlucky, indeed. Janet must do less than five more than three times the number of chores her sister must do. Janet’s sister must only do more than one more than the negative number of chores her sister does.

(a) Graph the system

(a) Write a system of inequalities to model the problem

(b) Shade the solution region

(b) Solve the inequality by graphing

(c) Check a solution from the solution region by substituting it back into the system of inequalities, verify that it yields a true inequality in both cases (d) Check a point from outside the solution region, verify that it yields a false inequality for at least one of the equations in the system.

Solutions Systems of Inequalities – 2009 By the end of this activity, you will be able to solve systems of inequalities, and apply them to real world situations. 1. Make a sketch to represent an example system of equations, label the boundary lines, and solution region. b

b

4. Solve each system of inequalities by shading the solution region, if no solution region exists, say so. (a) 2. y ≥ 4x + 2 x
2x + y > 5

(a) Show that points A, B, and C are not solutions to the system using substitution (b)

C (−4, −10)

A (2, 2)

y <2−x

y >x−2

(2) ≥ 4(2) + 2

(−10) ≥ 4(−4) + 2

2 ≥ 10

FALSE

(2) < (2) + 5

−10 ≥ −14

y ≤x−4 TRUE

No solution region. The regions do not overlap...

(−4) < (−10) + 5

2<7

TRUE

−4 < −5

FALSE

B (4, −5)

(c)

(−5) ≥ 4(4) + 2 −5 ≥ 18

FALSE

(4) < (−5) + 5 4<0

FALSE

3. With the system of inequalities below y > 2x − 3 y <7−x

5. Janet’s unfair mathematical parents.

(a) (b) (c) Check a solution from the solution region, Points may vary Check (−3, 0) (0) > 2(−3) − 3 0 > −9 (0) < 7 − (−3) 0 < 10

x≤3 y≥4

(a) Write a system of inequalities to model the problem, x number of chores done by Janet’s sister, y number of chores done by Janet. y < 3x + 5, and x > −y + 1, also x ≥ 0 and y ≥ 0 (b) Solve the inequality by graphing,

TRUE TRUE

(d) Check a point from outside the solution region Points may vary Check (3, 0) (0) > 2(3) − 3 0>3 (0) < 7 − (3) 0<4

FALSE TRUE

Notice that negative numbers of chores do not make sense, so exclude them from the solutions