Alg A: Capstone 2, Part 1 Review

Alg A: Capstone 2, Part 1 Review

By Allison Gilbert

​This quizizz will review...​

​1) Solving equations

2) Solving inequalities

3) Solving absolute value equations

4) Solving proportio​ns

​Please be prepared to watch the video, write down examples, and solve the problems that follow with a calculator.

Steps to Solve

• D - Distribute

• C - Combine Like Terms

• M - Move the variable to one side

• A - Get rid of adding or subtracting

• M - Get rid of multiplication or division

Solving Equations with Variables on Both Sides

Vocabulary

Identity- An equation that is true for every possible value of the variable.

-You can replace the variable with any value and the expressions on each side are equivalent.

Example: x+1=x+1

No Solution-

An equation has no solution if there is no value of the variable that makes the equation true.

Example- x+1=x+2

Learning Objective-

I can solve equations with variables on both sides to identify equations that are identities or have no solution.

Focus Question

How can you solve equations with variables on both sides?

Answer

You can use the properties of equality and inverse operations to write a series of simpler equivalent equations.

Steps:
-Write the original equation.
-Subtract 2x from each side. The coefficient 2 is less than the coefficient 5, so it is easier to subtract.
-Simplify. Now there is only one variable term. 
-Subtract 2 from each side.
-Simplify

GOAL!!!

Get the variable on one side of the equation and the numeric terms on the other side!

What is the solution of the equation? Is the equation an identity, or does it not have a solution?

How do you know?

Multiple Choice

28 + 3r = 10r

• 4

• 7

• 20

• 28

Multiple Choice

30 + 6t = 11t

• 3
• 5
• 6
• 30

Multiple Choice

7x = 3x + 24

• 6
• 10
• 14
• 4

Multiple Choice

What is the first step to solve this equation:
11 - 3x = 44

• Add 3 to both sides
• Subtract 11 from both sides
• Add 11 to both sides
• Divide 3 on both sides.

Multiple Choice

Solve:
3c + 5 = 23

• c = 3
• c = 6
• c = 7
• c = 18

Multiple Choice

Solve the equation:
7x - 4 = 3 + 8x

• x = 7
• x = -7
• x = 2/7
• x = 11

Multiple Choice

Solve for x:
5x - 14 = 8x + 4

• x = 6
• x = -6
• x = -18/13
• x = 10/3

Solving Inequalities

Please watch the video on the next slide to review solving inequalities

Multiple Choice

What is the inequality sign for greater than?

• $$<$$

• $$>$$

• $$\le$$

• $$\ge$$

Solving Inequalities

When solving inequalities the steps are the same as solving a regular equation, only difference is if you divide or multiply by a negative number the inequality sign flips directions.

Multiple Choice

If you divide an inequality with a less than sign by -3, what inequality sign would it change to?

• $$>$$

• $$<$$

• $$\ge$$

• $$\le$$

Solving Inequality Steps

• Get rid of all Parentheses

• Combine like terms

• Move all x terms to the left hand side (LHS)

• Move all constants to the right hand side (RHS)

• Get rid of the x’s coefficient by doing the opposite operations (multiplication -> division or division -> multiplication)

• BE CAREFUL WHEN DIVIDING/MULTIPLYING BY A NEGATIVE

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

When looking at this inequality what should be your first step? $$-1-6x-6>-11-7x$$  

• Get rid of parentheses

• move x terms to the LHS

• combine like terms

• move the constant to the RHS

Example 2

- 1 - 6x - 6 > - 11 - 7x

(-1 - 6) - 6x > -11 - 7x

-7 - 6x > -11 - 7x

-7 - 6x + 7x > -11 - 7x + 7x

-7 + x > -11

-7 + x + 7 > -11 + 7

x > -4

Multiple Choice

You try!

Solve. $$1-3x\ +6x>10-6x$$  

• x > 1

• x < -1

• x > 9

• x < 9

Multiple Choice

When looking at this inequality what should be your first step? $$3(1-2x)>3-6x$$  

• Get rid of parentheses

• move x terms to the LHS

• combine like terms

• move the constant to the RHS

Multiple Choice

Level 1

Match the graph with its inequality.

• 11 < a

• 11 > a

• 11 ≤ a

• 11 ≥ a

Multiple Choice

Level 1

Write an inequality for the statement:

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

• h < 48

• h > 48

• h ≤ 48

• h ≥ 48

Multiple Choice

Level 2

-4 + y < -8

• y < 4

• y < -12

• y < -4

• y < 12

Multiple Choice

Level 2

The room can hold a maximum of 400 people. If there are already 134 people there, how many more can come? Identify the inequality that represents this situation.

• x - 134 < 400

• x + 134 < 400

• x - 134 ≤ 400

• x + 134 ≤ 400

Multiple Choice

Level 2

23 < f + 12

• f > 11

• f > 35

• f < 35

• f < 11

Multiple Choice

-4 + y < -8

• y < 4
• y < -12
• y < -4
• y < 12

Multiple Choice

Does the given value make the inequality true?  x + 9 > 21, when x = 15

• True
• Not True

Multiple Choice

Solve

• x < -96
• x < 20
• x = 6
• x > 96

Multiple Choice

Does the sign need to flip when solving?
8y < (-40)

• YES
• NO

Multiple Choice

- 2k > 18

• k is greater than 20
• k is less than negative 9
• k is greater than negative 9
• k is less than 36 

Multiple Choice

-8x < 48

• x < -6
• x > -6
• x < 6
• x > 6

Multiple Choice

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

• ≤, ≥ 
• <, >

Multiple Choice

Does the given value make the inequality true? x + 9 > 21, when x = 15

• True

• Not True

Multiple Choice

Solve 

• a≤1
• a≤25
• a>1
• a<25

Definition

A compound inequality is made up of two inequalities connected by the word “AND” or the word “OR.”

Multiple Select

Which of the following inequalities are compound?

• 5 < 4x - 7 < 56

• 3 (x - 9) > 65

• 2x+ 5 < 67 - 3x

• x > 5 or x < -22

AND

Here is a video on how to solve a Compound Inequality that uses the word AND.

Multiple Choice

Which graph represents the answer to the compound inequality:  $$4\le x+2\le8$$  ?

Correct Answer

Multiple Choice

Which graph represents the answer to the compound inequality:  $$-9<x-10<5$$  ?

Correct Answer

Multiple Choice

Which graph represents the answer to the compound inequality:  $$-50<7x+6<-8$$  ?

Correct Answer

This one requires two steps.

OR

Here is a video on how to solve a Compound Inequality that uses the word OR.

Multiple Choice

Which graph represents the answer to the compound inequality:  $$2+r<12$$  or  $$r+5>19$$  

Correct Answer

Multiple Choice

Which graph represents the answer to the compound inequality:  $$7x\ge21$$  or  $$2x\le-2$$  

Correct Answer

Multiple Choice

Which graph represents the answer to the compound inequality:  $$-5x+2>27$$  or  $$x-3>2$$  

Multiple Choice

Which graph represents the answer to the compound inequality:  $$-18<3x-6\le-3$$  

Absolute value |a|

Absolute value represents a number's distance from zero on a number line. Because it represents a distance, an absolute value is positive.

|-2| = 2 Because -2 is two units away from zero on a number line.

|7| = 7 Because 7 is seven units away from zero on a number line.

Evaluating with Absolute Value

When evaluating mathematic expressions with absolute value, treat the absolute value bars as a grouping symbol, similar to parentheses.

2 -5|3 + 1| Add the numbers inside the bars first. Take the absolute value of that sum. Then multiply by five and subtract that product from 2.

2-5|4|--->2-5(4)--->2-20--->-18

Fill in the Blank

Evaluate |-3|+|-4+1|-|7-2|

Fill in the Blank

Evaluate 2+6|3-5|-9

Solving absolute value equations

When solving absolute value equations, it is important to understand the definition of absolute value.

|x| = 5

When solving the above equation, remember that absolute value is a number's distance from zero on the number line. If |x|=5, what number or number is five units away from zero on the number line? That could be 5 or -5. The equation has two answers: x = 5 and x = -5.

Multiple Choice

What is the reason that absolute value is always written as a positive?

• Absolute value is talking about numbers so it must be positive.
• Absolute value does not always have to be positive.
• Absolute value is like a clock it has only positive numbers.
• Absolute value is talking about distance, distance is always measured by positive numbers.

Multiple Choice

∣ 2x + 9 ∣ = 15

• {3}

• { -6,3}

• { -12,3}

• { -12,6}

Multiple Choice

-|−2r − 1| = 11

• {-6,5}
• {5,-6}
• -6
• No Solution

Multiple Choice

|−4 + 5x| = 16

• {16/5,12/5}
• {4,-5}
• {-12/5, 4}
• 4

Multiple Choice

∣ 2x + 9 ∣ = 15

• x = 3 
• x = 3
  x = -6
• x = 3
  x = -12
• x = 6
  x = -12

Multiple Select

$$\frac{x}{8}=\frac{15}{24}$$    1st: Multiply x(24), 2nd: Multipliy 8(15), 3rd: Solve for x.

• x=5

• x=120

• x=24

Multiple Select

6.   $$\frac{40}{y}=\frac{4}{5}$$  

• y=10

• y=50

• y=4

Multiple Select

7.   $$\frac{6}{2}=\frac{n}{14}$$  

• n=40

• n=41

• n=42

Multiple Select

8.   $$\frac{4}{10}=\frac{32}{k}$$  

• k=80

• k=320

• k=40

Multiple Select

10.   $$\frac{15}{21}=\frac{10}{e}$$  

• e=210

• e=14

• e=15

Multiple Select

12.   $$\frac{6}{4}=\frac{21}{r}$$  

• r=14

• r=12

• r=6