Regents Review #4

Inequalities

and

Systems Roslyn Middle SchoolResearch Honors Integrated Algebra

Simple Inequalities

1) Solve inequalities like you would solve an equation (use inverse operations to isolate the variable)

2) When multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign

3) Graph the solution set on a number line

Simple Inequalities

-3x – 4 > 8

-3x > 12

x < - 4

3(2x – 1) + 3x 4(2x + 1)

6x – 3 + 3x 8x + 4

9x – 3 8x + 4

x 7 -7 -6 -5 -4 -3 -2 -1

4 5 6 7 8 9 10

Simple InequalitiesWords to Symbols

At Least

Minimum

Cannot Exceed

At Most

Maximum

Example

In order to go to the movies, Connie and Stan need to put all their money together. Connie has three times as much as Stan. Together, they have more than $17. What is the least amount of money each of them can have?

Let x = Stan’s money Let 3x = Connie’s money

x + 3x > 17

4x > 17 x > 4.25

Since Stan has to have more than $4.25, the least amount of money he can have is $4.26.Since Connie has to have three times as much as Stan, the least amount that she can have is $12.78 (3 X 4.26).

Compound Inequalities

A compound inequality is a sentence with two inequality statements joined either by the word “OR” or by the word “AND”

“AND” Graph the solutions that both inequalities have in common

“OR” Graph the combination of both solutions sets

Compound Inequalities“AND”

5 > x > 9

x < 5 and x > 9

{ } or

2 3 4 5 6 7 8 9 10 11 12

-8 2x + 4 < x

-8 2x + 4 and 2x + 4 < x -12 2x and 4 < -x -6 x and -4 > x

-9 -8 -7 -6 -5 -4 -3 -2 -1 0

Compound Inequalities“OR”

x < -4 or x 6

-10 -8 -6 -4 -2 0 2 4 6 8 10

2x + 5 > 11 or 3x < 152x > 6 or x < 5 x > 3

x > 3 or x < 5

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

Linear Inequalities

Graph Linear Inequalities the same way you graph Linear Equations but…

1)Use a dashed line (----) if the signs are < or >

2)Use a solid line ( ) if the signs are or

3)Shade above the line if the signs are > or

4)Shade below the line if the signs are < or

Linear Inequalities

Graph -2y > 2x – 4

-2y > 2x – 4

y < - x + 2

m = b = 2 (0,2)Test point (0,0) -2y > 2x – 4 -2(0) > 2(0) – 4 0 > 0 – 4 0 > - 4 True

1

1

1

1

or

-2y > 2x - 4

Systems

A "system" of equations is a collection of equations in the same variable

When solving Linear Systems, there are three types of outcomes…

No Solution

y = 2x + 5y = 2x – 4

One Solution

y = -2x + 4y = 3x - 2

Infinite Solutions

y = 2x + 33y = 6x + 9

Systems

There are two ways to solve a Linear System

1)Graphically-graph both lines and determine the common solution (point of intersection)

2)Algebraically-Substitution Method-Elimination Method

SystemsSolving Systems Algebraically (Substitution)

x + y = 7 3x = 17 + y

Finding y

3x = 17 + y

3(7 – y) = 17 + y

21 – 3y = 17 + y

-4y = -4

y = 1

Finding x

x + y = 7

x + 1 = 7

x = 6

Solution (6,1)

x = 7 – y

Check

x + y = 76 + 1 = 7 7 = 7

3x = 17 + y3(6) = 17 + 1 18 = 18

Systems

Solving Systems Algebraically (Elimination)

5x – 2y = 10

2x + y = 31

5x – 2y = 10

2[2x + y = 31]

5x – 2y = 10

4x + 2y = 62+

9x + 0y = 72 9x = 72 x = 8

Finding y

2x + y = 312(8) + y = 31 16 + y = 31 y = 15

Solution (8, 15)

Check

5x – 2y = 105(8) – 2(15) = 10 40 – 30 = 10 10 = 10

4x + 2y = 624(8) + 2(15) = 6232 + 30 = 62 62 = 62

Systems

Using Systems to Solve Word ProblemsA discount movie theater charges $5 for an adult ticket and $2 for a child’s ticket. One Saturday, the theater sold 785 tickets for $3280. How many children’s tickets were sold?

Let x = the number of adult ticketsLet y = the number of children tickets

5x + 2y = 3280 x + y = 785

5x + 2y = 3280-5[x + y = 785]

5x + 2y = 3280 -5x – 5y = -3925+

0x – 3y = -645 -3y = -645 y = 215

Finding x

x + y = 785x + 215 = 785 x = 570

570 adult tickets215 children tickets

Systems

Solving Linear-Quadratic Systems Graphically

Two Solutions No SolutionOne Solution

Systems

Solving Linear-Quadratic Systems Graphically

y = x2 – 4x – 2 y = x – 2

y = x – 2 m =

b = -2 (0,-2)

y = x2 – 4x – 2 x = 2

2

4

)1(2

)4(

2

a

b

1

1 x y

-1 3

0 -2

1 -5

2 -6

3 -5

4 -2

5 3

Solutions (0,-2) and (5,3)

Systems

Solving Systems of Linear Inequalities

y < 3x m = 3/1 b = 0 (0,0)

y -2x + 3 m = -2/1 b = 3 (0,3)

Sy < 3xy -2x + 3

y < 3xy -2x + 3

1) Graph each inequality2) Label each inequality3) Label the solution region with S4) Label the solution region with the

system of inequalities

Regents Review #4

Now it’s time to study!

Using the information from this power point and your review packet,

complete the practice problems.