Factoring
Distributing
Combining Like Terms
Inequalities
You have Got to be kidding
100
Lin missed math class on the day they worked on expanding and factoring. Kiran is helping Lin catch up. Lin understands that expanding is using the distributive property, but she doesn’t understand what factoring is or why it works. How can Kiran explain factoring to Lin?
Factoring is the distributive property in the other direction. Instead of expanding a product to a sum of terms, factoring takes a sum of terms and makes it into a product by looking for common factors in the terms that can can be written outside the parentheses.
100
True or false. How do you know?
-3(6x-5y)= -18x+15y
True because -3 times 6x is -18x and -3 times -5y is +15y
I can't combine -18x and 15y because they are unlike terms so this is as far as I can take it.
100
True or false. How do you know?
5x−(8−6x)=−x−8
False
5x−(8−6x)= 5x-8+6x = 5x+6x -8= 11x-8
100
Which numberline is correct -3x-15>-24 true?
A
100
28 students travel on a field trip. The school has a van that can seat 12 students. The rest of the students will ride in cars, with 3 students in each car. Elena wonders if she should use the inequality
12+3n>28 or 12+3n ≥ 28 to figure out how many cars are needed. Kiran doesn’t think it matters in this case.
Do you agree with Kiran? Why or Why not
Agree with Kiran. In this case n represents a number of cars, so only whole number values of n make sense for the situation. (There can’t be fractions of cars or a negative number of cars.) 12+3n=28 has the solution n=16/3 so the number of cars needed is a minimum of 6.
200
Write an equivalent expression by factoring -24x+56
-8(3x-7) or 8(-3x+7) most accepted
-4(6 - 14). 4(-6x + 14)
-2(12x - 28). 2(-12x + 28) also true as are others
200
-1/3 (6x-15y)
negative one third times the difference of 6x and 15y
-2x + 5y
200
3-(5x-2)+9x
3-(5x-2)+9x
3 - 5x +2 + 9x
3 + 2 - 5x + 9x
5 + 4x or 4x + 5
200
Draw a number line which shows all the values of x that make the inequality true -3x -5 < 22.
x > -9
open circle at -9 with an arrow pointing towards 0
200
Solve for j 2/5(j+40)=−4
two fifths times the sum of j and 40 equals negative 4
j = -50
2/5 (j + 40)=-4
2/5j+16 =-4 (distribute) or j+40=-10 (divide first)
2/5j = -20 j+40-40=-10-40
j=-50
300
36a−16
4(9a-4) or 2(18a-8)
300
5-(x+7) What understood number is between - and (
1
We can think of the expression as 5-1(x+7)
300
Select all the equivalent expressions to −3(x+5)−4x+7
A. −3x−15−4x+7
B. −3x−4x−15+7
C. −7x−8
D. −3x−4x−12
E. −7x−12
A, B, C
300
Emma has read 20 pages of her book this week. For the rest of the week, she plans to read the same number of pages each day. There are 5 days left in the week. Before the week ends, Emma needs to read at least 50 pages to reach her goal.
Write an inequality for the situation.
5x + 20 ≥ 50
5 represents the days
x how many pages to read each day
20 how many pages previously read
50 minimum pages to read
≥ need to read exactly or more
300
In a ski lodge, the hot chocolate machine fills cups 6.2 ounces at a time. The machine turns off when 10 or less ounces of hot chocolate remain. The machine can hold a total of 112 ounces.
The expression 112-6.2c helps explain this situation.
What does the variable c stand for?
How many cups of hot chocolate can be dispensed before the machine turns off
400
Write an equivalent expression by factoring:
−4xy−12xz+20xw
4x(−y−3z+5w) or -4x(y+3z-5w)
400
-1/4(−8x+12y)
negative one fourth times the sum or negative 8x and 12y
2x-3y
400
Priya is simplifying the expression −2(6−3x)−5−x
Here is her work:
−2(6−3x)−5−x
−12−3x−5−x
−17−3x−x
−17−4x
Priya's work is incorrect. Which step did she make a mistake. Explain or show why this expression is not equivalent to −2(6−3x)−5−x.
Priya didn't distribute correctly. The second line should read -12+6x-5-x
Next combine like terms 6x-x -12-5
5x-17
Can also use substitution to show they aren't equivalent.
400
Elena pushes the elevator button, but the elevator does not move. The weight limit for the elevator is 450 pounds, but Elena and her load of identical packages weigh a total of 620 pounds. Each package weighs 37 pounds. How many boxes should she remove?
Write and solve an inequality to represent the situation.
620-37x ≤ 450
620 - 620 - 37x ≤ 450 -620
-37x ≤ -170
-37x/-37 ≤ -170/-37
x ≥ 4.595 This means they'd have to take off more than 4 boxes to meet weight requirements.
400
Write an inequality for the following graph:
x ≤ 25
x is less than or equal to 25
500
-30x+20y-55z
Ryan wrote the following as an equivalent expression
5(6x-4y+11z)
What is his mistake?
He should have factored a -5 or it would have been 5(-6x+4y-11z)
500
Expand 5a[3x-(-7y)+8z]
multiply the numbers, keep both variables
15ax-(-35ay)+40az or 15ax+35ay+40az
500
Tyler is simplifying the expression 6x−6+4x-5. Tyler’s work is shown below. It is incorrect. Explain the error or errors he made.
6x−6+4x-5
x+4x-5
5x-5
x
Tyler combined unlike terms. You can't subtract 6 from 6x and 5x-5 isn't x. He needed to use the communitive property to rearrange the expression.
6x+4x -6 -5 and could use associative property to group. (6x+4x) + (-6-5) = 10x -11 would be an equivalent expression
500
Deep sea spider crabs live at depths of -150 meters or lower. A submarine sets out to study the crabs in their natural habitat. It begins at -20 meters and descends steadily at a rate of 12 meters per second.
Write and solve an inequality that represents t, the time in seconds, until the submarine is lower than -150 meters.
-20 + -12t ≤ -150
-20 + 20 + -12t ≤ -150 + 20
-12t ≤ -130 (need to decend another 130 meters)
-12t/-12 ≤ -130/-12
t ≥ 10 5/6 It will take 10 5/6 seconds to get to a depth of -150 meters but we need to go deeper so any number greater than 10 5/6 will be part of the answer set.
500
Elena is trying to create a playlist that lasts no more than 2 hours (120 minutes). She has already added songs that total 15 minutes. She reads that the average song length on her music streaming service is 3.5 minutes. Elena writes the inequality 3.5x+15≥120 and solves it to find the solution x≥30
Explain how you know Elena made a mistake based on her solution.Fix Elena’s inequality and explain what each part of the inequality means.
Elena used the wrong inequality signs saying she needed 2 or more than 2 hours of music. The correct inequality is 3.5x+15≤120. The number 3.5 represents the average length of each song. The variable x represents the number of songs that Elena adds. The 15 represents the 15 minutes of songs that are already on the playlist. The ≤120 represents that the total number of minutes has to be less than or equal to 120 minutes.