View
2
Download
0
Category
Preview:
Citation preview
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 1 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
First Name: ________________________ Last Name: ________________________ Block: ______
Ch. 8 & 9 – Systems of Equations and Inequalities Notes
CH. 8.1 – SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY 2
Ch. 8.1 HW: p. 435 # 4, 5, 8, 20 4
8.2 – SOLVING SYSTEMS OF EQUATIONS ALGEBRAICALLY 5
Ch. 8.2 HW: p. 451 #3 – 5, 9, 10 10
9.0 – PRE-REQUISITES: WRITING INEQUALITIES (DOMAIN AND RANGE) 11
9.1 – LINEAR INEQUALITIES IN TWO VARIABLES 15
Ch. 9.1 HW: p. 472 #1 – 9 18
9.2 – QUADRATIC INEQUALITIES IN ONE VARIABLE 19
Ch. 9.2 HW: p. 484 # 1 – 9, 10 22
9.3 – QUADRATIC INEQUALITIES IN TWO VARIABLES 23
Ch. 9.3 HW: p. 496 # 1 – 8 (odd letters), 11 24
9.4 – GRAPHING AND SOLVING SYSTEMS OF LINEAR INEQUALITIES 25
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 2 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Ch. 8.1 – Solving Systems of Equations Graphically Definitions:
• A system of linear equations (with 2 variables) contains a pair of linear equations
• Solution to a linear system (in two variables) is any ordered pairs (x, y) that satisfy both equations.
There are several ways to solve linear systems:
•
•
• Examples: Systems of linear equations. 1) Solve the following linear system graphically.
a) x – y = -2 4x + 2y = 16
b) 2y – 3x = -2 3y + x = -3
Note:
• A system of equations does not have to be linear.
• Also, it can have more than 2 equations.
• To solve a system of equations with _____ number of variables require at least _____ equations.
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 3 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Examples: Systems of non-linear equations. 3) Solve the following system graphically.
a) y = x
2x – y = 6
b) y = x2 y = 4
4) Solve the following system graphically.
a)
0382
034
2=+−+
=+−
yxx
yx
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 4 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
b)
1182
35162
2
2
−=−−
−=−−
yxx
yxx
5) Using a graphing calculator, verify your solutions in question 4.
Ch. 8.1 HW: p. 435 # 4, 5, 8, 20
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 5 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
8.2 – Solving Systems of Equations Algebraically Definitions:
• Solutions to a system (in two variables) are any ordered pairs (x, y) that satisfy both equations. Solving a System of Linear-Linear Equations:
2) Solve the system of equations. 8=+ yx
1423 =− yx
In grade 10, you learnt how to solve such system by substitution or elimination. Use both methods to see which method you prefer.
Solve By Substitution: Isolate one of the variables: Which variable would be easiest to isolate? ________ From which equation? ________ Substitute the expression for _____ into the other equation.
Now that we’ve solved for _______, we need to solve for the other variable ______. It doesn’t matter which equation you choose. Choose the equation that will require least amount of work.
Solutions: ____________. Verify the solutions:
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 6 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Solve By Elimination: 8=+ yx
1423 =− yx
Which method would you prefer?
3) Solve the system of equations. Which method would be easier? xy 2=
1143 =+ yx
4) Solve the system of equations. Which method would be easier?
645
832
−=−
=+
yx
yx
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 7 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Solving a System of Linear-Quadratic Equations:
5) Solve the system of equations.
342 2=+− yxx
724 −=− yx
Solve By Substitution: Isolate one of the variables: Which variable would be easiest to isolate? ________ From which equation? ________
Substitute the expression for _____ into the other equation.
Now that we’ve solved for _______, we need to solve for the other variable ______. It doesn’t matter which equation you choose. Choose the equation that will require least amount of work.
Solutions: ____________. Verify the solutions:
342 2=+− yxx
724 −=− yx
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 8 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
342 2=+− yxx
724 −=− yx
Solve By Elimination: Rearrange the equations such that all the like terms are vertically aligned. Eliminate one of the variables. Which variable would be easiest to remove? ________. Now that we’ve solved for _______, we need to solve for the other variable ______. It doesn’t matter which equation you choose. Choose the equation that will require least amount of work. Solutions: ____________. Verify the solutions: Which method do you prefer? Substitution or elimination?
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 9 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
6) Solve the system of equations. 93 −=+ yx
94 2−=+− yxx
By Substitution:
By Elimination:
93 −=+ yx
94 2−=+− yxx
Which method do you prefer?
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 10 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
System of Quadratic-Quadratic Equations
7) Solve the system of equations.
1982+=+ xyx
1172−=− xyx
8) Solve the system of equations.
16 2−=−− yxx
644 2−=−− yxx
Ch. 8.2 HW: p. 451 #3 – 5, 9, 10
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 11 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
9.0 – Pre-requisites: Writing Inequalities (domain and range) Recap:
Sign Read as Example Read as
>
≥
<
≤
Domain Write the Inequality to describe the domain
Another way to write the domain
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 12 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Multiple Choice Questions:
1) Which of the following inequalities represents:
a) 2−≥x b) 2−>x c) 2−≤x d) 2−<x
2) Which of the following inequalities represents:
a) 1≥x b) 1>x c) 1≤x d) 1<x
3) Which of the following inequalities represents:
a) 3−>x and 4<x b) 43 <<− x
c) 3−<x or 4>x d) both a and b
4) Which of the following inequalities represents:
a) 2−≥x and 1<x b) 21 −≥≥ x
c) 2−≤x or 1>x d) both b and c
5) What is the domain of the relation below:
a) 24 ≤<− x b) 24 ≤≤− x
c) 24 ≥>− x d) 24 ≥≥− x
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 13 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
6) Which of the following are also functions?
a) III only b) I and III only
c) II and IV only d) I, III, and IV only
7) What is the domain of the relation below:
a) (-4, 2] b) [-4, 2)
c) (-4, 2) d) [-4, 2]
8) Determine the range of the relation below:
a) 2<y b) 2≥y
c) 2≤y d) 4−≥y
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 14 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
9) Determine the domain of the relation below:
I. All x values between 2 and 6 inclusive. II. (2, 6) III. [2, 6]
IV. 62 ≤≤ x
V. 51 << x
a) I, III, IV b) I only
c) II, V only d) V only
10) Determine the domain of the relation below:
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 15 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
9.1 – Linear Inequalities in Two Variables
Examples 1: Graph each inequality
y ≥ 2x + 1
y > 32
3−x
Examples 1: Graph each inequality
y ≤ 53 −x
y < x
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 16 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Examples 2: Write an inequality to represent each graph
Examples 3: Graph each inequality
y – 2x ≥ 1
4x + 3y > -12
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 17 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Examples 4: Graph each inequality on a coordinate grid.
y 4≤
1−>x
Examples 4: Graph each inequality on a number line.
y 4≤
1−>x
Examples: Multiple Choice Questions
1. Replace the = with either >, ≥ , < or ≤ to represent the inequality graphed below:
2x + 2y = 6
You can assume a solid line: a) 2x + 2y ≤ 6 b) 2x + 2y ≥ 6 c) 2x + 2y > 6 d) 2x + 2y < 6
2. Replace the = with either >, ≥ , < or ≤ to represent the inequality graphed below:
3x - y = 4
You can assume a solid line: a) 3x - y < 4 b) 3x - y > 4 c) 3x - y ≤ 4 d) 3x - y ≥ 4
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 18 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
3. Which of the following ordered pairs are solutions to the given inequality:
a) {(0, 0), (-1, 3), (-3, 3), (5,5)} b) {(0, -3), (0, -1), (1, -1), (2, -1)} c) {(3, 10), (-1, 2), (3, 4), (5, 5)} d) {(-5, -1), (2, 2), (0, -10), (2, 2)}
332 =− yx
Ch. 9.1 HW: p. 472 #1 – 9
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 19 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
9.2 – Quadratic Inequalities in One Variable
1. Given the graph of 432−−= xxy below,
Solve 0432>−− xx
2. Given the graph of 432−−= xxy below,
Solve 0432≤−− xx
3. Given the graph of )(xfy = below,
a) Solve 0)( <xf
b) Solve 0)( ≥xf
4. Given the graph of )(xfy = below,
a) Solve 0)( <xf
b) Solve 0)( ≥xf
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 20 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
5. Given the graph of )(xfy = below,
a) Solve 0)( >xf
b) Solve 0)( ≥xf
c) Solve 0)( <xf
d) Solve 0)( ≤xf
6. Given the graph of )3)(1()( +−= xxxf and 5=y
a) Solve 5)( >xf
b) Solve 5)( ≤xf
Solving Quadratic Inequalities that are factorable:
1) Solve 0)3)(1( ≥+− xx
2) Solve 0)1)(2(2 >−+− xx
3) Solve 0322<−− xx
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 21 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
4) Solve 016102≤+− xx
5) Solve 0122<++− xx
Solving Quadratic Inequalities that are NOT factorable:
6) Solve 01272 2≥−− xx
Step 1: First solve for the zero(es) of the quadratic function, 1272 2−−= xxy .
Step 2: Sketch the quadratic function. Solve the inequality.
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 22 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
7) Solve 1042>− xx
8) Solve 432 2−≥+− xx
Ch. 9.2 HW: p. 484 # 1 – 9, 10
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 23 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
9.3 – Quadratic Inequalities in Two Variables
When graphing inequalities:
• Use ______________ line for > or < inequality symbols.
• Use ______________ line for ≥ or ≤ inequality symbols.
• Shade _____________ the curve when curvey > or curvey ≥ .
• Shade _____________ the curve when curvey < or curvey ≤ .
Examples:
1. Graph each inequality.
1)3(2 2+−−≥ xy .
6)1(3 2−+< xy
2)1(
2
1−> xy
43
1 2+−≤ xy
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 24 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
2. Graph each inequality.
422++−≥ xxy
342 2−+> xxy
52
1 2−+≤ xxy
123
1 2++−> xxy
Ch. 9.3 HW: p. 496 # 1 – 8 (odd letters), 11
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 25 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
9.4 – Graphing and Solving Systems of Linear Inequalities
A solution to a system of linear inequalities in two variables is a set of ordered pairs that satisfies all the inequalities in the system.
Example1: Solve the system of inequalities by graphing.
3x + 2y ≤ 6 4x – 3y > 12
xy2
1>
62
3+−≤ xy
x ≥ 1
Example 2: Word Problems involving inequalities
A company makes pencils and pens. Due to staffing limitation, no more than 400 pencils and up to 500 pens can be made in one day. Due to supply limitation, no more than 600 writing utensils can be made in a day. The company sells one pencil for $0.50 and one pen for $1.00 Determine how
many pens and pencils should be made in a day to maximize sales.
Let x = # of pencils made in a day y = # of pens made in a day.
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 26 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Practice Questions: Solve the system of inequalities by graphing.
y < 3 y ≥ x – 3 2x + y < 5
2x + 2y < 4 3x – y ≥ 1
y > 2 x ≤ 3 y < 2x + 10
y < 2x – 3 y > 2x + 1
Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities
Created by Ms. Lee 27 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11
Word Problems:
Sarah likes to swim and jog. She burns about 500 Cal/h swimming and 600 Cal/h jogging. It costs $5/h to swim at a local swimming pool. She is willing to spend less than $20 per week exercising. Due to her knee injury, she cannot jog more than 5 hours a week. She wants to burn at least 3000 Calories per week exercising. a) Clearly write all inequalities.
b) Graph the inequalities and clearly indicate the feasible solution.
c) Can she swim for 5 hours and jog for 3 hours? Why or why not?
Let x = # of hours Sarah swims y = # of hours Sarah jogs
Recommended