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Math 1 Module 4: Equations & Inequalities

4.1 Ready, Set, Go!

Questions 1 - 6

This video shows you how to draw a line on a graph when the equation is in line-intercept form: https://www.youtube.com/watch?v=AQ0tktNv9GY

 

This video shows you how to determine if a point lies on a line or not: https://www.youtube.com/watch?v=SUPRzFGYDs8

Question 7

To see if the expressions are equivalent, simplify them both as much as you can. In this case, you need to expand the parentheses in the first expression to "distribute" the minus sign. This video has a similar example: https://www.youtube.com/watch?v=0wV7QcPvdQ0

Question 8

To see if the expressions are equivalent, simplify them both as much as you can. In this case, you need to expand the parentheses in the second expression to "distribute" the 2 in front, like this: https://www.youtube.com/watch?v=v-6MShC82ow

Questions 9 - 10

Simplify the first expression to see if it's equivalent to the second expression.

Question 11

To equations have the same solutions if the equations are equal to each other. Simplify the first equation and if it's the same as the second equation, then they will also have the same solution.

Question 12

I recommend simplifying all the expressions as must as possible to see which ones are equal to each other, like the first example in this video: https://www.youtube.com/watch?v=5CR9rRrsp9o

Question 13

You have to solve a linear equation with a fraction. This video shows you a similar example: https://www.youtube.com/watch?v=wjSbFruEwWI

Question 14

First, you need to eliminate the parentheses and simplify the fraction. Then following the fraction rule to find x: https://www.youtube.com/watch?v=GYNK6NDNEFk

Question 15

You have to solve a linear equation with a fraction. This video shows you a similar example: https://www.youtube.com/watch?v=wjSbFruEwWI

Question 16

You need to substitute a = -3 to the equation, evaluate the left side then compare with the right side. Remember to include minus sign if necessary.

Question 17

You need to substitute x = 4/3 to the equation, evaluate the left side then compare with the right side. Remember to include minus sign if necessary.

Question 18

You need to substitute y = 2 to the equation, evaluate the left side then compare with the right side. Remember to include minus sign if necessary.

Question 19

You need to substitute z = -5 to the equation, evaluate the left side and the right side then compare them. Remember to include minus sign if necessary.

Question 20

You need to substitute w = 1/4 to the equation, evaluate the left side and the right side then compare them. Remember to include minus sign if necessary.

Question 21

You need to substitute b = 5 to the equation, evaluate the left side and the right side then compare them. Remember to include minus sign if necessary. This is more tricky question which you need to be careful whether there's any b in you equation.

4.2 Ready, Set, Go!

Questions 1 - 10

This video reviews inverse operations: https://www.youtube.com/watch?v=QQoPAU5L52s

Questions 11 - 17

Start by determining the units for each variable in the expression. This video shows you how to write out the meaning of the expression using units: https://www.youtube.com/watch?v=CSNeYAV95UU

Questions 18 - 24

This video has several examples of converting regular notation into set notation and interval notation. I think it would be helpful with this question if you don't remember how the notation works: https://www.youtube.com/watch?v=wmX1PI-Rs-8

Question 25

This video works through several examples of converting inequalities to set notation and interval notation: https://www.youtube.com/watch?v=wmX1PI-Rs-8

This video introduces the whole numbers: https://www.youtube.com/watch?v=0myyT8H3yro

Questions 26 - 31

This video works through several examples where they graph inequalities on a number line: https://www.youtube.com/watch?v=wmX1PI-Rs-8

4.3 Ready, Set, Go!

Questions 1 - 7

In the example, they take the inequality -9 < 2 ("-9 is less than 2") which is true and apply the operation + 3 to both sides. When you do that, you get -6 < 5. This inequality is still true, and that's the answer for in the last column of the example table.

For these questions, you do the same thing as the example, but instead of adding 3 to both sides, you add/subtract the appropriate amount from both sides and then figure out if the inequality is still true.

Question 8
This video covers all the situations were the inequality sign flips: https://www.youtube.com/watch?v=v1tp-BrL84g

Question 9

This video shows you how to solve a similar literal equation for a different variable: https://www.youtube.com/watch?v=_KRBOMwkIBE

Question 10

This video shows you how to solve a similar literal equation: https://www.youtube.com/watch?v=_KRBOMwkIBE

Questions 11 - 12

You have to solve this literal equation for a different variable. This video shows you how to do a similar problem: https://www.youtube.com/watch?v=5xcMQlshSJM

Questions 13 - 14

This video shows you how to solve a similar multistep literal equation: https://www.youtube.com/watch?v=lCeMVbx798M

Questions 15 - 18

This video shows you how to solve a similar rational literal equation for a different variable: https://www.youtube.com/watch?v=EbzHG1mdj54

Questions 19 - 20

This video shows you how to solve a similar multistep literal equation: https://www.youtube.com/watch?v=lCeMVbx798M

Questions 21 - 23

The x-intercept is the point (x,y) where y = 0 in the table of data. Similarly, the y-intercept is the point where x = 0. You answer should be two points for the x- and y-intercepts.

Questions 24 - 25

The x-intercept is the point (x,y) where y = 0 on the graph. Similarly, the y-intercept is the point where x = 0. You answer should be two points for the x- and y-intercepts.

 

This video shows you how to find the x- and y-intercepts from a graph: https://www.youtube.com/watch?v=SP6VjMIivW0

Questions 26 - 28

This problem requires you to solve the equation for x use inversion operations, this video shows you a similar example: https://www.youtube.com/watch?v=QQoPAU5L52s

Questions 29 - 31

This problem requires you to solve the equation for x. Note that x is on both sides of the equal sign, this video shows you a similar example: https://www.youtube.com/watch?v=9PqygqYo0ck

Question 32

This problem requires you to solve the equation for X. This is actually just a single step problem - divid both sides by (B + 7) and you'll isolate X!

4.4 Ready, Set, Go!

Question 1

We are told that Virginia charges a $10 fee per job plus $0.20 per square foot of paint and that one job total cost was $50. We need to figure out how many square feet that job was. The way to do this is to set up and solve an equation where x represents the variable that we don't know (in this case the "number of square feet of paint"). Start by filling in the blanks Total Cost = $___ + $___x

When you have the equation for Total Cost, we can figure out how many feet were in a $50 job. You need to set the expression for Total Cost = $50 and solve for x.

Questions 2

We are told that renting the ice skating rink costs $200 plus $4 per person and that one bill was $324. We need to figure out how many people attended. The way to do this is to set up and solve an equation where x represents the variable that we don't know (in this case the "number of attendees"). Start by filling in the blanks Total Cost = $___ + $___x

When you have the equation for Total Cost, we can figure out how many people attended if the bill was $324. You need to set the expression for Total Cost = $324 and solve for x.

Question 3

Try this out by picking a value for x that is greater than 12 (to satisfy the first inequality) and then see if the same value for x works in the second inequality.

Question 4

In this situation, you are only expanding the parentheses on one side of the inequality, you aren't applying a transformation to both sides of the inequality, so the direction of the inequality shouldn't change.

Questions 5 - 7

This video contains short summary of operations on inequalities: https://www.youtube.com/watch?v=WHMU1KVjtA0

Pay attention to the direction of the inequality sign here, should it change when you subtract from both sides?

Questions 8 - 11

Start by solving the inequality for x. Your answer will be in the form x < A (where A is a number you calculate). This video walks through a similar example: https://www.youtube.com/watch?v=ONuIIvgkE3c

Next you need to see if the specified value of x is part of the "solution set" which just means "is the value of x a possible solution?". A solution is valid if it satisfies x < A.

Questions 12 - 14
Start by solving the inequality for x. This video walks through solving basic inequalities: https://www.youtube.com/watch?v=ONuIIvgkE3c

After you've solved the inequality, you need to graph it on the number line. This video has lots of examples of graphing inequalities: https://www.youtube.com/watch?v=nif2PKA9bXA

Questions 15 - 17

Start by solving the inequality for x. This example shows you how to solve inequalities with fractions: https://www.youtube.com/watch?v=8i_o4plUBTo

After you've solved the inequality, you need to graph it on the number line. This video has lots of examples of graphing inequalities: https://www.youtube.com/watch?v=nif2PKA9bXA

Questions 18 - 20

The 2nd example in this video (starting at 1:30) shows you how to solve multistep inequalities for x: https://youtu.be/xOxvyeSl0uA?t=90

 

Questions 21 - 26

In this question, you need to solve a system of linear equations by substitution. This video goes through a similar examples: https://www.youtube.com/watch?v=V7H1oUHXPkg

4.5 Ready, Set, Go!

Questions 1 - 10

Start by turning "I am thinking of a number that" into "x". Then you have "x is greater than 13". Write that inequality statement in mathematical notation.

This video goes through the inequality signs: https://www.youtube.com/watch?v=0j9DMsg4bvk

Question 11

The equation for average of two values is Average = (value1 + value2)/2 and we know that the average must be GREATER than 875, which means (value1 + value2)/2 > 875. If value1 = 903, what is value2? Solve this equation algebraically.

This video reviews interval and set notation: https://www.youtube.com/watch?v=wmX1PI-Rs-8

If you need more help with writing inequalities from contexts, this video has a full lesson: https://www.youtube.com/watch?v=TwV60jkZ5EY

Question 12

The equation for average of three values is Average = (value1 + value2 + value3)/3 and we know that the average must be at least 93, which means (value1 + value2 + value3)/3 ≥ 93. If value1 = 89 and value2 = 94, what is value3? Solve this equation algebraically.

This video reviews interval and set notation: https://www.youtube.com/watch?v=wmX1PI-Rs-8

If you need more help with writing inequalities from contexts, this video has a full lesson: https://www.youtube.com/watch?v=TwV60jkZ5EY

Question 13

For the second company, Cost2 = $19.99 + $0.25x, where x is the number of messages sent. If the cost of the first company doesn't depend on how many texts are sent, what is the equation for Cost1 for them?

When you have both equations, choose your favorite systems of equations method (substitution, elimination, etc) to find out when the second company starts being more expensive. This video has a refresher on solving systems of equations by substitution: https://www.youtube.com/watch?v=V7H1oUHXPkg

This video reviews interval and set notation: https://www.youtube.com/watch?v=wmX1PI-Rs-8

Question 21 - 24

This video shows you how to solve systems of equations by substitution: https://www.youtube.com/watch?v=V7H1oUHXPkg

4.6 Ready, Set, Go!

Question 1

For part (a) Darren starts with $100 and spends $50 on a membership. How much does Darren have left? How many admissions tickets ($15) can Darren buy with what's left?

Part b) if Darren doesn't buy the membership, there is $100 to spend on $25 visits. How many times can he go?

For parts c) and d), you need to do the same calculations as part a) and b) but with a new budget of $160. What do you notice? What do you wonder?

Question 2

For part a) we know that there are 20 questions and 100 total points available, which means each question is worth 100/20 points (leave a comment if you need help with this step!). The equation calculates Jade's score, so what information can you use to do that? Hint: You can set x = "the number of questions Jade got right" because we don't know how many Jade got we have to leave it as an unknown.

Part b) divide 70 by the number of points per question to see how many questions she needs to get right to earn 70 points. You'll notice that your answer is a decimal. You'll need to round it up to the nearest whole number because you can't have part of a question.

For part c), do the same thing as part b) but with 83 points instead of 70.

Finally, for part d) you know that there are 100 points total, so 60% is the same as 60 points. In the retest she gets those 60 points again and half of the rest of the points she missed.

Questions 3 - 8

In this problem, you need to solve the inequality for x. The first five minutes of this video have similar examples: https://www.youtube.com/watch?v=0X-bMeIN53I

Questions 9 - 10

Simplify the inequality by getting all x terms on the left side and all numbers combined on the right side. Then, follow this video for help graphing the inequality on the number line: https://www.youtube.com/watch?v=wmX1PI-Rs-8

Questions 11 - 14

In this problem, you need to solve the multistep inequality for x. This video has similar examples: https://www.youtube.com/watch?v=0X-bMeIN53I

Questions 15 - 17

This video shows you how to solve a similar literal equation for a variable: https://www.youtube.com/watch?v=_KRBOMwkIBE

Question 18

This video shows you how to rearrange the variables in a literal equation to isolate a particular variable: https://www.youtube.com/watch?v=5xcMQlshSJM

Questions 19 - 20

This video shows you how to rearrange the variables in a literal equation to isolate a particular variable: https://www.youtube.com/watch?v=_KRBOMwkIBE

Questions 21 - 23

This video shows you how to draw a linear graph from a linear equation (you'll need to convert one of the equations to point slope form first): https://www.youtube.com/watch?v=kgD48XXVT1c

4.7 Ready, Set, Go!

Questions 1 - 2

This video shows you how to fill the missing data in a similar table: https://www.youtube.com/watch?v=cMa02EEAFC8

Question 3

The equation for this table is in reciprocal form. When x = 1, we are told h(1) = -13. We can then find h(2) = h(2-1) + 6 = h(1) + 6 = -13 + 6 = -7. Repeat the process to find h(3) and another several times to find h(15).

Question 4

The equation for this table is in reciprocal form. When x = 1, we are told d(1) = 34. We can then find d(2) = d(2-1) - 2 = d(1) + 6 = ... (you can do the rest!). Repeat the process several times to find d(10).

Questions 5 - 6

This video shows you how to fill the missing data in a similar table: https://www.youtube.com/watch?v=cMa02EEAFC8

Questions 7 - 10

Let x, y, z be bags of chips, gallons of drinks, and dozen of cookies accordingly. You can create a table that represent x, y, z then fill numbers in with information from the context

Questions 11 - 16

This video has a great overview of the differences between geometric and arithmetic sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

This video shows you how to write explicit and geometric equations for arithmetic (linear) and geometric (exponential) sequences: https://www.youtube.com/watch?v=mcNGhrvp52s

4.8 Ready, Set, Go!

Question 1

For these types of questions, I recommend focusing on the x term first. What happened to turn x into 3x? Then look at the y term, did the same transformation apply there?

Questions 2

For these types of questions, I recommend focusing on the x term first. What happened to turn 4x into x? Then look at the y term, did the same transformation apply there?

Questions 3

Start by getting all the terms in the two equations in the same format. For the equation y = -(3/2)x + 5, you'll have to move the x-term to the left side of the equals sign. This is the first transformation.

When the equations are in the same format I recommend focusing on the x term first. What happened to turn 6x into (3/2)x? Then look at the y-term, did the same transformation apply there?

Questions 4 - 5

The goal is to figure out if the equations are the same. For lines to be the same, all the terms have to have the same coefficients OR all the coefficients of the first equation need to be a multiple of the coefficients in the second equation. 

 

Notice that the first equation has a minus in front of the y-term but the second equation doesn't.

Math fact: if two lines are "equivalent" it means they are parallel. If they are not equivalent, they'll intersect at some point.

Question 6

The goal is to figure out if the equations are the same. Start by getting all the terms in the two equations in the same format. For the equation y = (2/5)x + 10, you'll have to move the x-term to the left side of the equals sign.

For lines to be the same, all the terms have to have the same coefficients OR all the coefficients of the first equation need to be a multiple of the coefficients in the second equation.

Math fact: if two lines are "equivalent" it means they are parallel. If they are not equivalent, they'll intersect at some point.

 

Questions 7 - 10

This video has a lot of similar examples of solving linear equations: https://www.youtube.com/watch?v=GmMX3-nTWbE

Questions 11 - 12

To find Total Cost matrix (2x2), you need to multiply the Equipment Matrix (2x3) with the Cost Matrix (3x2). This video works through a similar example: https://www.youtube.com/watch?v=OMA2Mwo0aZg

Question 13

Each little blue triangle has a width of 1 unit and the width of the large triangle changes with the step number. In step 1 there is just 1 little triangle. In step two, there are two little triangles along the base, so the base is 2 units. In step three, there are three little triangles along the base, so the base is 3 units.

Question 14

For this question, we need to describe the relationship between the step number and the number of little blue triangles. I recommend you start by listing out the information you're given in a table (e.g. Step 1 -> 1 triangle, Step 2 -> 3 triangles, Step 3 -> 6 triangles).

Question 15

To figure out how the perimeter of the big triangle changes with step number, I recommend putting the data you are given into a table. For step one, the three sides are each 1 unit, so the total perimeter is 3. In step 2, the three sides are each 2 units, so the total perimeter is 6. Find the perimeter for Step 3 and see if you can put the pattern into words.

Question 16

The blue and white triangles are all equilateral which means all their corners are 60 degree angles. I've put this little diagram together to show you what they mean for Step 1 and Step 2. Do the same for Step 3 and then put all the data into a table and see if you can put the pattern you find into words.

Question 17

To figure out how the number of white triangles changes with step number, I recommend putting the data you are given into a table. For step one, there are no white triangles. For step 2, there is one. Fill out the table and up to Step 5 and see if you can put the pattern into words.

4.9 Ready, Set, Go!

Questions 1 - 5

This video shows you how to draw a linear graph from a linear equation: https://www.youtube.com/watch?v=kgD48XXVT1c


This video shows you how to draw a linear graph from a table of linear data: https://www.youtube.com/watch?v=L5kDZ7iejkA

Question 6

The first two examples in this video show you how to add 2x3 matrices: https://www.youtube.com/watch?v=WR9qCSXJlyY

Question 7

The third example in this video (starting at 3:06) shows you how to subtract 2x2 matrices: https://www.youtube.com/watch?v=WR9qCSXJlyY

This video is also pretty good: https://www.youtube.com/watch?v=7jb_AO_hRc8

Question 8

This video shows you how to multiply matrices by a scalar: https://www.youtube.com/watch?v=TbaltFbJ3wE

Question 9
This video shows you how to add matrices: https://www.youtube.com/watch?v=WR9qCSXJlyY

Questions 10 - 11

This video shows you how to multiply matrices by a scalar: https://www.youtube.com/watch?v=TbaltFbJ3wE

Questions 12 - 13
This video shows you how to multiply two matrices together: https://www.youtube.com/watch?v=kT4Mp9EdVqs

Questions 14 - 19

Substitute x = 7, y = -3, and z = 5 into the function, then follow order of operations to evaluate the expression.

If you need a refresher on order of operations, this video goes through all the rules: https://www.youtube.com/watch?v=dAgfnK528RA

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