8.18 Inequalities

Things to Remember:

-The inequalities that contains the equal signs are closed circles when modeling on the number line.

-The inequalities without the equal sign are open circles when modeling on the number line.

Multiple Choice

Decide if the inequality is open circle or closed circle: 4x > 16

• Open Circle

• Closed Circle

• Flip the sign

• I don't know

Graphing Inequalities

• When graphing your inequality on the number line, the open circle means the number is not included.

• When graphing your inequality on the number line, the closed circle means the number is included.

​VOCABULARY

​What an inequality MEANS in an algebraic context.

​Given: x - 3 = 7

​If we solve it, x = 10

​

​Which means we plug it in

​​10 - 3 = 7

​7 = 7

​

​7 = 7 is a true statement

​So x is 10 and ONLY 10

​Given: x - 3 > 7

​If we solve it, x > 10

​

​x has to be bigger than 10

​So let's pick 15 and plug it in

​15 - 3 > 7

​15 - 3 > 7

​12 > 7

​

12 > 7 is a true statement

​So x is ANY NUMBER BIGGER THAN 10

Multiple Select

Given x > 3, choose all numbers that could be x.

• 3

• -3

• 12

• 4

• 1,895

​What an inequality MEANS continued

x = 5: "x equals 5" so x is 5

​

x < 5: "x is less than 5" so x is 4, 3, 2, 1 ,0, -1, -2 ...

​

x ≤ 5: "x is less than or equal to 5" so x is 5, 4, 3, 2, 1 ,0, -1, -2 ...

​

x > 5: "x is greater than 5" so x is 6, 7, 8, 9, 10, 11, 12 ...

​

x ≥ 5: "x is greater than or equal to 5" so x is 5, 6, 7, 8, 9, 10, 11, 12 ..

Multiple Select

Given x ≤ -7, choose all numbers that could be x.

• -7

• -9

• -6

• -1

• -129

​VOCABULARY

​GRAPH: the collection of all points whose coordinates satisfy a given relation.

​Example: x = 6

​

​

​

​

​Example: x < 6

​EXAMPLE: x ≥ -4

• ​ ​meaning all numbers bigger than four AND including four

• so on the graph, we will use a CLOSED CIRCLE​

• ​Graphically is mean 6 is part fo the solution set or part of the answers

​

​

​EXAMPLE: x > -4 

Draw

Graph: x ≥ -1

Multiple Choice

Which graph represents

x < 3

Graphing Inequalities examples for reference

Multiple Choice

Which graph represents

x ≥ -17

​What if the variable is on the right?

​Circle rules are the same.

​You may want to rewrite it but it has to 'eat' the same number or variable.

​

​Example: 2 < x 

Draw

Graph: -2 > x

Multiple Select

Which inequalities represent the graph?

• x < -4

• x > -4

• -4 > x

• -4 < x

Multiple Choice

Which situation can be represented by the inequality

$$h\ \ge\ 45$$  ?

• You must do more than 45 minutes of homework tonight

• I've been watching TV for 45 minutes.

• You can eat no more than 45 M&M's

• You must be at least 45 inches tall to ride the rollercoaster.

Review...

Multiple Choice

How would you graph the following inequality:

$$c\ \le\ -4$$  

• open circle on -4, arrow to the right

• open circle on -4, arrow to the left

• closed circle on -4, arrow to the right

• closed circle on -4, arrow to the left

Multiple Choice

CJ had $500 in his account at the beginning of the summer. He wants to have at least $200 left in his account by the end of summer. He withdraws $25 each week for food, clothes, and games. Which inequality represents CJ's situation?

• $$500\ +\ 25x\ \le\ 200$$

• $$500\ -\ 25x\ \ge\ 200$$

• $$500\ -\ 25x\ \le\ 200$$

• $$500\ +\ 25x\ >\ 200$$

Multiple Choice

Greater than or equal to

• <

• >

• ≤

• ≥

Multiple Choice

Greater than

• <

• >

• ≤

• ≥

Multiple Choice

Less than or equal to

• <

• >

• ≤

• ≥

Multiple Choice

When graphing an inequality, you use an OPEN dot when you use which symbol?

• <, >

• ≤, ≥ 

• ≤, <

• ≥, >

Multiple Choice

When you graph an inequality, you used a CLOSED dot when you use which symbols?

• ≤, ≥ 

• <, >

• ≤, <

• ≥, >

Multiple Choice

Match the graph with its inequality.

• a < 11

• a > 11

• a ≥ 11

• a ≤ 11

Multiple Choice

Which inequality is represented by the following graph

• x < -8

• x > -8

• x ≥ -8

• x ≤ -8

Multiple Choice

When graphing an inequality on the number line the OPEN circle means the number is______?

• Not included

• Included

Try to solve this on paper!

What would you do first?

What would you do second?

WHAT MUST YOU DO?

Multiple Choice

You flip an inequality symbol when you...

• subtract

• multiply and divide only

• multiple by a negative number

• multiple or divide by a negative number

Multiple Choice

Solve and choose the correct graph:

• A

• B

• C

• D

Multiple Choice

-4x + 14 ≤ 54

• x ≥ - 10

• x ≤ -10

• x ≥ 10

• x ≤ 10

Multiple Choice

What is this symbol?

• less than

• greater than

• less than or equal to

• greater than or equal to

Example 1

- 8x + 2x - 16 < - 5x + 7x

(-8x + 2x) - 16 < (-5x + 7x)

-6x - 16 < 2x

-6x - 16 - 2x < 2x - 2x

-8x - 16 < 0

-8x - 16 + 16 < 0 + 16

-8x < 16

-8x/-8 < 16/-8

x > -2

Multiple Choice

$$Solve:\ -7\left(n-6\right)<91$$  

• $$n<7$$  

• $$n>7$$  

• $$n<-7$$  

• $$n>-7$$  

Multiple Choice

$$\frac{3}{4}x+5>11$$  

• $$x>8$$  

• $$x>-8$$  

• $$x>-1$$  

• $$x<4$$  

Multiple Choice

Do I need to flip my sign?
-3x  ≥ 12

• Yes

• No

Multiple Choice

-2x + 3(-4x + 6) ≤116

• x ≤ -7

• x ≥ 29

• x ≥ -7

• x ≤ 29

Multiple Choice

$$−5(2𝑥+2)+5x>15$$  

• $$x<-5$$  

• $$x<5$$  

• $$x>5$$  

• $$x>-5$$  

Multiple Choice

3x - 5 > -26

• x < -7

• x > -7

• x > 7

• x < 7

Multiple Choice

6(n + 2) < -72

• n < -14

• n < 35/3

• n < -10

• n > -14

Multiple Choice

4(2a + 3) < 3(a - 1)

• a < -3

• a > -3

• a < 3

• a > 3

Multiple Choice

Solve:
6x+10>5x-12

• x>2

• x<2

• x<-22

• x>-22