Math Exam 1

< and > means...

Dotted line

Infinity or negative infinity

use parentheses ( or )

Solve the equation. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) |2x − 24.8| = 32.4

1.For this equation you do the same thing, drop the absolute value signs and solve 2.Then drop the absolute value signs and solve but put a negative in front of 32.4

Let C = {-4, -1, 0, 2, 4} and D = {-4, 2, 4, 6, 8}. Find C ∪ D. C Unions D

{-4, -1, 0, 2, 4, 6, 8} All the points in common

Let C = {−4, −1, 0, 2, 4} and D = {−4, 2, 4, 6, 8}. C ∩ D... C intersects D

{−4, 2, 4} Only the points in common

Solve the compound inequality. Write the solution set in interval notation. 4x + 1 < 3x − 2 and x/2 + 9 ≤ 7

1.Solve both equations 2.Multiply second equation by 2 to get rid of the fraction 3.End up with -3 and -4 going to - infinity 4.This means that you have to shade from -4 to infinity to satisfy the two numbers

Factor the sum of cubes. 729t^3 + u^3

1. Use the formula for factoring the sum of cubes a³ + b³= (a+b) (a²-ab+b²) 2. Follow the formula: -What number cubed equals 729t^3 ? -What number cubed equals u^3 ? 3. (9t + u) (9^2 - 9tu - u^2) 4.(9t + u) (81 - 9tu - u^2) is your answer

Solve the double inequality. Graph the solution set and write it using interval notation. 0.9 < 2x − 0.7 < 1.5

1.Add .7 to both sides 2.Divide by 2 to both sides 3.Plot your points 4. (.8,1.1) is your answer

Graph the solutions of the system of inequalities, when possible. (4.5) x − y ≤ 4 x + 2y ≤ 4 x ≥ 0

1.Do the same thing as before, except now there is a 3rd equation. 2.Make sure when shading it is shared by all 3 equations 3. The x ≥ 0 goes right through the y axis and is shaded to the right

Factor, if possible. (If not possible, enter the original expression.) 9y^2 − 49

1.Factor the 9, y and the 49 2.Take 3 out and y out, and take 7 out of 49 3.(3y-9) (3y+9) is your answer

Simplify the rational expression. Thats a divide sign 6x^2 − 7x − 3 ------------- 3x^2 + 7x + 2

1.Factor the top equation: (3x + 1) (2x - 3) 2.Then factor the bottom equation: (x + 2) (3x + 1) 3.Cancel common factor : (3x + 1) 4.2x-3/x+2 is your answer

Simplify the rational expression. m^3 − mn^2 ------------------------ mn^2 + 6m^2n − 7m^3

1.Factor the top equation: m(m^2 - n^2) 2.Factor the bottom equation: m(-7m^2 +6mn +n^2) 3.Cancel the common factor from the top to the bottom equation: cancel "m" 4.You now have: m^2 - n^2 on the top equation and -7m^2 +6mn +n^2 on the bottom 5. Factor the top and the bottom equations TOP: (m + n) (m - n) BOTTOM: (n - m) (7m +n) 6.Finally cancel the common factor (m - n) 7.ANSWER: -m + n / 7m + n

Solve the equation. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x/9 = 6

1.First multiply by 9 to get rid of the fraction 2.You will get 54 3.Then you get -54 when you add a negative in front of it

Solve the equation. (Enter your answers as a comma-separated list.) |4x + 2| = |8 − 2x|

1.For this equation, drop the absolute value signs and solve to get your first solution 2.Then for the other solution, you drop the absolute value signs and put the '8-2x' in parentheses and put a negative in front of it: -(8-2x) 3.Solve the equation as this and find your second solution

Factor, if possible. (If not possible, enter the original expression.) (8.7) x^2 − 81

1.For this you factor 2.(x-9) (x+9) 3.Make sure you have one positive and one negative

Graph the solution of the system. (4.5) x ≥ 1 y ≤ 1

1.Horizontal and vertical lines at 1 on both axis

Solve the compound inequality. Write the solution set (if one exists) in interval notation and then graph it. x ≤ -2 OR x > 7

1.Plot both lines, they are already solved 2.Since it says OR you will use 'U' or union 3.Your answer in interval notation is: (-Infinity,-2] U (7, infinity)

Steps to graphing a solution to the system (4.5) x + 2y ≤ 6 2x − y ≥ 2

1.Set x equal to zero and solve the first equation 2.Set y equal to zero and solve the first equation 3.Plot points 4.Do the same for the other equation 5.Shade in coordinate that both lines share

Solve the inequality. Graph the solution set. |x| < 5

1.Since the absolute value sign means both positive and negative answers, you graph (-5,5)

Solve the compound inequality. Graph the solution set (if one exists) and write it using interval notation. (8.4) 5x - 5 > 5 and x + 4 ≤ 7

1.Solve both equations 2.Since it says AND, you must solve and include both equations 3. Use ( or ) for < and >.....and use [ and ] for ≤ or ≥ 4.Your answer is (2,3]

Solve the double inequality. Write the solution set in interval notation and then graph it. 3 ≤ x + 1 ≤ 5

1.Subtract 1 from each side 2.You will get 2 ≤ x ≤ 4 3. Your answer will be [2,4]

Factor completely. 256a^4 − 625b^4

1.Take out 16a^2 from 256a^4 2.Take out 25b^2 3.You will get (16a^2 - 25b^2) 4.Then you factor this 5.Take out 4a from 16a^2 6.Take out 5b from 25b^2 7.(4a-5b) (4a+5b) 8. Take (16a^2 - 25b^2) down with the other factored equations 9. (16a^2 - 25b^2) (4a - 5b) (4a + 5b) is your answer

Factor the expression. Factor out any GCF first. 2x^2 − 578

1.Take out 2 because it is the GCF of both 2x^2 and 578 2.Then take out 17 from 578 3.2 (x-17) (x+17) is your answer

Factor the difference of two squares. 9r^4 − 289s^2

1.Take out 3 r^2 from 9r^4 2.Take out 17s from 289s^2 3.(3r^2 - 17s) (3r^2 + 17s)

Factor, if possible. (If not possible, enter the original expression.) 64a^2 − 49b^2

1.Take out 8a and 9b 2.(8a-7b) (8a+7b)

Steps to graphing a solution the system (4.5) y ≥ x y ≤ 1/3 x + 2

1.The first equation goes through the origin 2.The second equation you have to mulitply by 3 to get rid of the fraction 3.Then find the coordinates for the second equation 4.Plot points and graph

Solve the equation. (Enter your answers as a comma-separated list.) −6|5x − 8| + 11 = 11

1.This is the same as before except all you do is subtract 11. 2.You will only get one solution for this answer because it will be equal to 0

Factor the difference of cubes. 512m^3 − x^6

1.Use the formula for factoring the difference of cubes: a^3 - b^3= (a-b) (a^2 - ab + b^2) 2. Follow the formula: -What number cubed equals 512m^3 ? -What number cubed equals x^6 ? 3. (8m - x^2) (8m^2 - 8mx^2 + x^2) 4. (8m - x^2) (64m^2 - 8mx^2 + x^4) this is your answer

Factor the sum of cubes. 512r^3 + s^3

1.Use the formula for factoring the sum of cubes a³ + b³= (a+b) (a²-ab+b²) 2.Follow the formula: -What number cubed equals 512r^3? -What number cubed equals s^3? 3.(8r + s) = (8r^2 - 8rs + s^2) 4.(8r + s) = (64r^2 - 8rs + s^2) This is your answer

Factor by first grouping the appropriate terms. x^2 - y^2 +x + y

1.We must take out x and y 2.(x+y) 3.Then add (x-y) 4.Then throw a 1 at the end 5.(x + y) (x - y + 1)

Solve the equation. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) |x − 2.9| = −16.3

1.When an absolute value sign equals a negative, it cannot exist. No solution

Solve the inequality. Graph the solution set. |7x − 2| < 12

1.When graphing this, you find both solutions just as before, and all you do it graph them 2.Your solutions will be (-10/7,2) 3.Graph accordingly

Solve the compound inequality. Write the solution set (if one exists) in interval notation and then graph it. -5(x + 5) ≥ 15 or 3x + 5 < 14

1.When you solve both equations your get x ≤ -8 and x < 3 2.This means you pick the number that they both cross. 3. 3 cannot be -8, so you have to start from 3 and go to negative infinity 4. (-infinity, 3) is your answer

Factor the polynomial. (p+q)^2 - r^2

1.Write (p+q) twice 2.(P+Q) (P+Q) 3.Then throw r into both equations 4.(P+Q+R) (P+Q-R) 5.Make sure to have both a positive and negative r

Find the domain of the rational function. Express your answer in interval notation. (8.8) f(x)= 5x/x+6

1.X cannot equal -6 2.So writing the interval equation you would include: (- infinity, -6) U (-6, positive infinity)

Steps to graphing a solution to the system (4.5) y ≤ 2x x + y < 1

1.Y is solved, so solve for x 2.Cannot use 0 because line runs through origin 3.Try 1, 2 (x,y) (1,2) That will be one point 4.Try -1, -2 (x,y) (-1,-2) that will be another 5.(0,0) will be a coordinate along with the others 6.Solve the other equation the same and graph

Factor by first grouping the appropriate terms. y^2 + 12y + 36 - z^2

1.You must add or subtract to get 12 2.You must multiply to get 36 3.You also must take y and z out when you factor 4.(y + 6 - z) (y + 6 + z) is your answer -Notice how 6 + 6 equals 12, the middle term -Notice how 6 times 6 equals 36 -Notice how z is both positive and negative

Solve the equation. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) |3x + 2| = 13

1.You must find 2 answers because of the absolute value, both a positive and negative value. 2.First, Drop the absolute value signs and then solve the equation as it is 3.Then, drop the absolute value signs and make the 13 negative and solve the equation. 4.Write your 2 solutions in a common separated list

≤ and ≥

Graphed with a CLOSED circle or a bracket

< and >

Graphed with an open circle or parentheses

≤ and ≥

Solid line

No solution means

The lines you plotted are in 2 places at once, it is impossible for it to be true.

(x,y)

Y is the vertical and x is the horizontal

Factoring the difference of squares

a^2 - b^2 = (a - b) (a + b)

Factoring the difference of cubes

a^3 - b^3= (a-b) (a^2- ab + b^2)

Factoring the sum of cubes

a³ + b³= (a+b) (a²-ab+b²)