Regents Review #4 Inequalities and Systems
Feb 25, 2016
Regents Review #4
Inequalities
and Systems
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 decide 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 three times as much as Stan, she has $12.78.
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”
-12 2x < -8
2x -12 and 2x < -8 x -6 and x < -4 -6 x < - 4
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
-3 < x 4
x > -3 and x 4
-4 -3 -2 -1 0 1 2 3 4 5 6
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 InequalitiesGraph 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
11
11
or
-2y > 2x - 4
SystemsA "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
Systems y = 4x – 1 m = b = -1 (0,-1)
3x + 2y = 20 2y = -3x + 20 y = - + 10
m = - b = 10 (0,10)
14
23
23
Solution (2,7)
3x + 2y = 20
Y =
4x –
1
Check (2, 7)
y = 4x – 1 3x + 2y = 207= 4(2) – 1 3(2) + 2(7) = 207 = 8 – 1 6 + 14 = 207 = 7 20 = 20
Solve the system y = 4x – 1 and 3x + 2y = 20 graphically
SystemsSolving Linear 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 Linear 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 Graphicallyy = x2 – 4x – 2 y = x – 2
y = x – 2 m =
b = -2 (0,-2)
y = x2 – 4x – 2 x = 2
24
)1(2)4(
2
ab
11 x y
-1 3
0 -2
1 -5
2 -6
3 -5
4 -2
5 3
Solutions (0,-2) and (5,3)
y = x2 – 4x – 2
y = x – 2
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
1) Graph each inequality2) Label each inequality3) Label the solution region with S
Solve the system: y < 3x y -2x + 3
Now it’s your turn to review on your own! Using the information presented today and your review
packet, complete the practice problems in the packet.
Regents Review #5 is
FRIDAY, May 31st BE THERE!!!!