Ch. 9 - Systems of Equations and Inequalities Notes AnsKey · Ch. 8 & 9 – Systems of Equations and Inequalities Notes CH. 8.1 – SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY 2 Ch

Pre-Calculus 11 Chapter 9 – Systems of Equations and Inequalities

Created by Ms. Lee 1 of 27 Reference: McGraw-Hill Ryerson Pre-Calculus 11

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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



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.



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



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



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:



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



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



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?



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?



System of Quadratic-Quadratic Equations


1982+=+ xyx

1172−=− xyx


16 2−=−− yxx

644 2−=−− yxx

Ch. 8.2 HW: p. 451 #3 – 5, 9, 10



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



Multiple Choice Questions:

1) Which of the following inequalities represents:

a) 2−≥x b) 2−>x c) 2−≤x d) 2−<x


a) 1≥x b) 1>x c) 1≤x d) 1<x


a) 3−>x and 4<x b) 43 <<− x

c) 3−<x or 4>x d) both a and b


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



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



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:



9.1 – Linear Inequalities in Two Variables

Examples 1: Graph each inequality

y ≥ 2x + 1

y > 32

3−x


y ≤ 53 −x

y < x



Examples 2: Write an inequality to represent each graph


y – 2x ≥ 1

4x + 3y > -12



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



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



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


a) Solve 0)( <xf

b) Solve 0)( ≥xf




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



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.



7) Solve 1042>− xx

8) Solve 432 2−≥+− xx

Ch. 9.2 HW: p. 484 # 1 – 9, 10



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



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



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.



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



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

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Ch. 9 - Systems of Equations and Inequalities Notes AnsKey · Ch. 8 & 9 – Systems of Equations and Inequalities Notes CH. 8.1 – SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY 2 Ch