Module 10 Summative Test Review

Graphing inequalities on a number line.

Notice the X>3 and X<3 have an open circle, that is because 3 is not included in the inequality. Notice that the X≥3 and X≤3 have a closed circle, that is because 3 is included in the inequality. Make sure you use the correct circle based on the inequality symbol. By placing the circle above the number line, it will be easier to tell if it is open or closed.

Solving One-Step Inequalities & Graphing Inequalities

Graphing Inequalities

• < and > have open circles

• ≤ and ≥ have closed circles

Graphing inequalities on a number line.

Notice the X>3 and X<3 have an open circle, that is because 3 is not included in the inequality. Notice that the X≥3 and X≤3 have a closed circle, that is because 3 is included in the inequality. Make sure you use the correct circle based on the inequality symbol. By placing the circle above the number line, it will be easier to tell if it is open or closed.

What does X > 3 actually mean?

Notice that our arrows on our number lines start at 3 and then go left or right depending on the inequality sign. This means that X could be any number the arrow covers, unlike equal signs which means X can only equal one number. X can not equal a number the arrow does not cover. Pro-tip: if X is on the left side of the inequality, then the arrow matches the direction of the inequality.

Multiple Choice

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

• <, >

• ≤, ≥ 

• ≤, <

• ≥, >

Multiple Choice

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

• ≤, ≥ 

• <, >

• ≤, <

• ≥, >

Multiple Choice

When graphing b < 6 would you use an open or closed circle?

• Open

• Closed

Multiple Choice

What inequality does the number line graph represent?

• x≥ 3

• x > 3

• x < -3

• x ≤ 3

Multiple Choice

Pick the correct inequality.

• A

• B

• C

• D

Multiple Choice

Match the graph with its inequality.

• 11 < a

• 11 > a

•  11 ≤ a

•  11 ≥ a

Multiple Choice

Match the graph with its inequality.

• b > –2

• b < –2

• b ≥ –2

• b ≤ –2

Multiple Choice

"At most" and "No more than" are both clue words for...

• <
• >
• ≤
• ≥

Multiple Choice

Match:
You must be at least 48 inches tall to ride the bumper cars.

• h < 48
• h > 48
• h ≤ 48
• h ≥ 48

Multiple Choice

Jim has $8 to spend on pizza. The slice is $4 and each topping is $0.50. Create an inequality to represent the number of toppings Jim can get if he only buys one slice of pizza.

• 4x + 0.50 ≥ 8
• .50x + 4 < 8
• 0.50x +4 ≤ 8
• 4x +0.50 ≤ 8

Multiple Choice

Juan is saving for a new computer and has saved $213. He is saving $3 a day and needs at least $654 for the computer he wants. How many weeks will it take him to save for the computer he wants?

• at most 147 weeks

• at most 21 weeks

• at least 147 weeks

• at least 21 weeks

Solving One-Step Inequalities

• Isolate the variable

• Use inverse operations to help you isolate the variable

x - 5 ≤ 3

• Since the operation in the inequality is subtraction, we will use addition to solve.

• Add 5 to both sides on the inequality sign.

• 5 +3=8

• x≤8

• If we were to graph this, the circle would be closed!

x + 4 ≥9

• Since the operation in the inequality is addition, we will use subtraction to solve.

• Subtract 4 from both sides on the inequality sign.

• 9-4=5

• x≥5

• If we were to graph this, the circle would be closed!

Multiple Choice

• d≥59

• d≤69

• d≥49

• d<58

Multiple Choice

• y<139

• y≥55

• y>55

• y≤139

Multiple Choice

• q≤46

• q<46

• q≥46

• q<46

6x ≤ 18

• Since the operation in the inequality is multiplication, we will use division to solve.

• Divide both sides of the inequality sign by 6

• 18 divided by 6 equals 3

• x≤3

• If we were to graph this, the circle would be closed!

• Since the operation in the inequality is division, we will use multiplication to solve.

• Multiply both sides of the inequality sign by 5.

• 4 times 5=20

• x>20

• If we were to graph this, the circle would be open!

Multiple Choice

• m>7

• m≤5

• 5>m

• 5<m

Multiple Choice

• x<4

• x≤1

• x≤4

• x≥1

Multiple Choice

• z≤1

• z>9

• z≤9

• z<9

Remember solving equations by isolating the variable?

We solve inequalities the same way! There can be one-step or two-step or multi-step depending on how many operations there are. Just remember to isolate the variable by getting the variable on one side of the inequality and everything else to the other side.

See this rule in action

Notice we are multiplying both sides by negative two. Notice how the greater than > symbol changed to a less than symbol <. Keep in mind that the variable can be on either side. X<6 is the same as 6>X. This is read X is less than 6, so generally we want the X on the left side, so be careful to keep the inequality facing the correct way if you change it.

Try some questions...

Use paper and pencil if needed.

Multiple Choice

Does the sign need to flip when solving?
-2x ≥ 22

• YES
• NO

Multiple Choice

Solve for X

- 4X ≤ 12

• X ≤ 3

• X ≥ 3

• X ≤ - 3

• X ≥ - 3

Multiple Choice

When you multiply or divide both sides by a negative number

in an inequality, you should

• Flip the fraction

• Change the direction of the inequality

• Eat a big breakfast

• Crocodiles have V shaped noses

Multiple Choice

Which picture best shows the answer to this inequality?

-2x ≤ - 4

Multiple Choice

Solve for x

4x + 3 ≤ −6

• x ≥ −9

• x ≤ −9

• x ≥ −2.25

• x ≤ −2.25

Multiple Choice

Solve for x

3x − 8 ≥ 4(x − 1.5)

• x ≥ 3.5

• x ≤ 3.5

• x ≥ −2

• x ≤ −2

Multiple Choice

Solve for x

6x − 7 > 2x + 17

• x > 6

• x < 6

• x > 7

• x < 7

Multiple Select

Which is the solution to the following inequality?

d - 5 > -4

-3

• d < 17

• d > 17

• d < 7

• d > -7

Lesson 8.1a: Solving Systems of Linear Equations by Graphing

CCSS—8.EE.8a

Open Ended

OPENING ASSESSMENT:

Tell whether each equation has one, zero, or infinitely many solutions.

1) 5(x - 3) + 6 = 5x - 9 ______________

2) 5(x - 3) + 6 = 5x -10 ______________

3) 5(x - 3) + 6 = 4x + 3 ______________

One Solution

• The two lines intersect (touch) at ONE POINT

• Your answer should be an ordered pair (x, y)

No Solution

• The two lines NEVER intersect (touch)

• 
  

Infinitely Many Solution

• The equations are on SAME LINE Your answer should be an ordered pair (x, y)

Multiple Choice

How many solutions will this system have?

• infinitely many solutions

• one solution

• no solution

• 2

Multiple Choice

How many solutions will this system have?

• no solution

• infinitely many solutions

• one solution

Multiple Choice

How many solutions will this system have? (Hint: There is a pink line and a blue line, one on top of the other.)

• infinitely many solutions

• one solution

• no solution

Fill in the Blanks

What is the solution?

** Write it as an ordered pair such as: (-x,y)

Multiple Choice

What is the solution?

• (1, -1)
• (-1, 1)
• (0, -2)
• (2, 0)

Multiple Choice

When you graph the exact same equation twice,

• you will have no solution. 
• you will have one solution.
• you will have infinite solutions. 
• you will graph a giraffe. 

Multiple Choice

How many solutions will this system have? 

• No solution
• One Solution
• I Don't Know
• Infinitely Many Solutions

Multiple Choice

How many solutions will this system have?

• One
• Two
• No Solution
• Infinitely Many Solutions

Multiple Choice

^{What is the solution?} 

• 1
• -2
• (1, 2)
• (1, -1)

Solving Systems of Linear Equations by Substitution

Step #1 Isolate a variable in one of the equations to get an expression

Solving Systems

Steps #2 Substitute that expression (from step #1) into the other equation.

Solving Systems

Step #3 Solve for the intersection point (x , y).

Solving Systems

step #4 write your answer as an order pair (x, y).

Multiple Choice

The solution to a system of equations is any ordered pair that makes both equations true. 

• TRUE
• FALSE

Multiple Choice

Solve this system by substitution.

• (1, -4)

• (1, 4)

• (-4, -7)

• (-7, -4)

Multiple Choice

Using Substitution, solve for the system of linear equations:

$$x=-3y+1$$  

$$-3x-3y=-15$$  

• Infinitely many solutions

• (-1,5)

• (7,-2)

• No solution

Multiple Choice

Using Substitution, solve for the following system of linear equations: $$5x-2y=18$$  

$$y=-2x+9$$  

• (4,1)

• No solution

• Infinitely many solutions

• (5,-2)

Solving Systems of Linear Equations using elimination

Multiple Choice

What is the additive inverse of 9?

• 9

• 0

• -9

• 18

Multiple Choice

What does 3 + (-3)=

• 6

• -6

• 0

• 3

Multiple Choice

What does 5x + (-5x) =

• 10x

• -10x

• 0

• 5x

Solving Linear Systems by Elimination

Sometimes we can add equations that have two variables to obtain a new equation that only has one variable. In this sense, we are ELIMINATING a variable in order to solve the system

Steps:

1. Add the equations to eliminate one variable
2. Solve the new equation
3. Substitute value found in step 2 into either original equation to find the value of the other variable

Multiple Choice

Solve by elimination:
4x+9y=28
-4x-y=-28

• (-7,0)
• (6,0)
• (-6,0)
• (7,0)

Multiple Choice

Solve by elimination:
-9x-4y=-20

5x+4y=4

• (-4,4)
• (4,4)
• (4,-4)
• (-4,-4)

Multiple Choice

Solve by elimination:
-x+2y=17

2x+2y=-10

• ARN
• (-9,4)
• (-9,-4)
• (9,-4)

Multiple Choice

Solve the system by multiplication.
3x + 2y = 16
7x + y = 19

• (-2,5)
• (-2,-5)
• (2,-5)
• (2,5)

Multiple Choice

Solve by elimination:
3x+7y=23

-3x-7y=-17

• No solution
• ARN
• (-3,3)
• (3,3)