Topic 1:
Real Number
Real Number
Simplify the expression below:
5b2 • 2b3 • 4b5
When multiplying common bases (they are all variables of b with exponents), you multiply the coefficients and add the exponents.
(5x2x4)b2+3+5
= 40b10
Simplify the following expression:
4x + 7 - x + 2
Combine like terms
(4 - 1)x + (7+2)
= 3x + 9
Frank records the ages, x, and heights, y, of the six students he tutors. Determine whether this relation represents a function?
(9, 54), (10, 54), (9, 61), (8, 45), (12, 65), (8, 50)
To be a function, every input must have exactly one output. This cannot represent a function because there is one input with two outputs. 9 has an output of 54 and 61.
What is the difference between a strong associate and a weak association?
In a strong association, data points are clustered up enough to form a linear line.
In a weak association, data points are more spaced out, and have visible outliers.
What is the solution to a system of linear functions?
The point (x, y) where the two lines intersect.
8.14114114... a rational number or irrational number?
This is an irrational number because it is non-terminating, cannot be made into a fraction because it does not have a repeating pattern.
Simplify the following expression:
-3(4x + 6) + 10
* Order of operations: PEMDAS
* Distributive Property: multiply -3 to 4x and 6
- 3 • 4x - 3 • 6
* Simplify and bring down the 10: -12x - 18 + 10
* Combine like terms: -12x + (-18 + 10)
* Simplify: -12x - 8
Draw a qualitative graph that describes a function of a hiker's altitude (in feet) on a hike over the course of 30 minutes.
The hiker started her hike up the mountain with a light jog. She then slowed down and walked for a few minutes. She then jogged again for a few minutes. She took a break for a good while to catch her breath. She then headed back down.
The y-axis should be the altitude of the hiker in feet. The x-axis should be the time in minutes.
The graph should show a sharp narrow increase, then a less narrow increase, then a sharp narrow increase, then a flat horizontal line, then a decrease going downhill.
How do you determine the line that best fits the trend of a scatter plot?
The line that goes through the middle of the data points.
What are the different kinds of solutions a system can have, and what does each solution mean?
1) no solution means the lines do not intersect because they are parallel
2) one solution means the lines intersect at one point
3) infinite solutions means the lines are the same, and intersect at every single point
A cube has a volume of 64 cubic inches. What is the length if one side?
The cube root of 64 is 4, because 4 multplied by its self 3 times is 64.
4 x 4 x 4 = 64
Each side length measures 4 inches.
Solve this equation:
-9(x + 6) = -207
*Distributive Property... -9 • x -9 • 6
*Simplify... -9x - 54 = -207
*Inverse Operation... -9x - 54 + 54 = -207 + 54
*Simplify... -9x = -153
*Inverse Operation... -9x/-9 = -153/-9
*Simplify... x = 17
What is the slope of the line that passes through the points (12, -18) and (11, 7)?
Slope Formula:
m = (7) - (-18) / (11) - (12)
m = -25 / 1
m = -25
Corey asked 82 to people at his school what type of fruit they prefer. Complete the frequency table.
Apples/ Grapes/ Oranges/ Total
Students 22 / ___ / 24 / ___
Teachers ____ / 4 / ____ / ___
Total 24 / ___ / 25 / 82
1) Teachers - Apples: 24 - 22 = 2
2) Teachers - Oranges: 25 - 24 = 1
3) Total - Teachers: 2 + 4 + 1 = 7
4) Total - Students: 82 - 7 = 75
5) Students - Grapes: 75 - 24 - 22 = 29
6) Total - Grapes: 29 + 4 = 33
Apples / Grapes / Oranges / Total
Students 22 / 29 / 24 / 75
Teachers 2 / 4 / 1 / 7
Total 24 / 33 / 25 / 82
How can you tell how many solutions a system has without looking at their graphs?
You can look at their slopes.
If they have equal slopes, they will have no solution. If they have different slopes, they will have one solution. If the slopes are equal and the y-intercept (equal equations), they will have infinite solutions.
Simplify 5.7777... as a fraction.
- set x = 5.7777
- count number of repeating digits... (1)
- multiply 10 to both sides... 10x = 57.777
- subtract x to both sides... 10x - x = 57.777 - 5.777
- simplify... 9x = 52.000
- solve for x... 9x/9 = 52/9
- x = 52/9
Solve this equation:
4x + 3 = x + 2(x+7)
1) Simplify both expressions on both sides of the equation (distributive property and combining like terms)
2) isolate the variable by using inverse operations
x = 11
Georgie opens a gym membership at TruFit. The fee to make the account is $55. Then, he will be charged $19.99 each month. Write an equation that represents the total cost,s y Georgie will pay for x number of months.
y = (slope/R.O.C.)x + y-intercept/ initial value
y = 19.99x + 55
At one point last year, the local animal shelter had only cats and dogs. There were 74 animals in all. Of the cats, 25 were male and 14 were female. Of the dogs, 23 were male and 12 were female.
Construct a two-way frequency table of the data:
Cat / Dog / Total
Male 25 / 23 / 48
Female 14 / 12 / 26
Total 39 / 35 / 74
How many solutions does the following system have?
2x - 6y = 30
y = 1/3x + 1
No solution. The slopes are equal, so the lines will be parallel and will not intersect.
y = 1/3x - 5
y = 1/3x + 1
8a15b8
------
2a4b3
When dividing common bases, you divide their coefficients and subtract their exponents.
(8/2)a15 - 4b8 - 3
= 4a11b5
Solve this equation:
-3(-2x - 5) = -2(-3x - 3)
This equation has no solution, because
6x + 15 = 6x + 5
The 6x's cancel out, and 15 can never be equal to 5.
What is the equation of the line that passes through (12, 29) and (8.5, 22)?
1. Use slope formula to find slope of the line
2. substitute slope into slope-intecerpt form
3. substitute y-value from ordered pair into equation in slope intercept form
4. solve for b, y-intercept
5. Substitute y-intercept into equation in slope intercept form
y = 3x + 4
Boys / Girls / Total
Red 18 / 22 / 40
Blue 10 / 14 / 24
Green 10 / 6 / 16
Total 38 / 42 / 80
Boys / Girls / Total
Red 47% / 52% / 50%
Blue 26% / 33% / 30%
Green 26% / 14% / 20%
Total 100% / 100% / 100%
How many solutions will this system have? How do you know?
2x - 3y = 6
4x - 6y = 12
Infinite solutions, because the lines are the same since the equations are equal.
y = 2/3x - 2
y = 2/3x - 2