Algebra EOC Practice Test #1 Notes

Question 1

Problem: George earns $20 a day plus $0.05 per flyer and wants to earn at least $65 a day.
Variable: Let x be the number of flyers distributed.
Inequality: The situation is represented by the inequality $20 + 0.05x \ge 65$ .

Question 2

Problem: Divide $(16x^6 - 12x^4 + 4x^2)$ by $4x^2$ .
Solution:
$\frac{16x^6 - 12x^4 + 4x^2}{4x^2} = \frac{16x^6}{4x^2} - \frac{12x^4}{4x^2} + \frac{4x^2}{4x^2} = 4x^4 - 3x^2 + 1$

Question 3

Problem: Solve the compound inequality $p + 1 < -1$ OR $p - 5 > 7$ .
Solution:
- p + 1 < -1 \Rightarrow p < -2
- p - 5 > 7 \Rightarrow p > 12
Graph: The graph represents the solutions $p < -2$ or $p > 12$ .

Question 4

Problem: Compare two job offers: ABC Company offers $45,000, and XYZ Company offers $38,000 plus a 2% commission on sales.
Variable: Let s be the amount of sales.
Inequality: We want to find when XYZ's salary is greater than ABC's salary: 38000 + 0.02s > 45000.
Solution:
0.02s > 7000 \Rightarrow s > \frac{7000}{0.02} \Rightarrow s > 350,000
Answer: The salary at XYZ Company is greater than the salary at ABC Company when s > 350,000.

Question 5

Problem: Solve $4s = -29$ .
Solution:
$4s = -29 \Rightarrow s = \frac{-29}{4} = -4.5$

Question 6

Problem: Paula needs an average of 80 or more to get a B. She got a 72 on her first test. What grade does she need on the second test?
Variable: Let x be the grade on the second test.
Inequality: The average of the two test scores must be greater than or equal to 80:
$\frac{72 + x}{2} \ge 80$
Solution:
$72 + x \ge 160 \Rightarrow x \ge 160 - 72 \Rightarrow x \ge 88$
Answer: Paula needs to get at least 88 on the second test.

Question 7

Problem: Find the equation of the line shown in the graph.
Analysis: The line has a y-intercept of -5. Visually determine the slope.
Answer: $y = \frac{3}{2}x - 5$

Question 8

Problem: Solve $m - 8 \le 14$ .
Solution:
$m - 8 \le 14 \Rightarrow m \le 14 + 8 \Rightarrow m \le 22$
Answer: $m \le 22$

Question 9

Problem: Graph the line with slope $\frac{1}{2}$ and y-intercept 3.
Analysis: The line passes through the point (0, 3) and has a slope of 1/2.

Question 10

Problem: Identify which relation is a function.
Definition: A function is a relation where each input (x-value) has exactly one output (y-value).
Analysis:
- a. {(-2, -2), (-2, -1), (-2, 0), (-2, 1), (-2, 2)} - Not a function because -2 is repeated with different outputs.
- b. {(1, 0), (-1, 0), (2, 1), (-2, 1), (3, 2), (-3, 2)} - A function, each x has a unique y.
- c. {(-2, 1), (-1, 2), (0, 0), (-1, 1), (2, -2)} - Not a function because -1 is repeated with different outputs.
- d. {(-3, 3), (1, 3), (-3, 2), (1, 2), (-3, 1), (1, 1)} - Not a function because -3 and 1 are repeated with different outputs.
Answer: b. {(1, 0), (-1, 0), (2, 1), (-2, 1), (3, 2), (-3, 2)}

Question 11

Problem: Simplify $(a^3b)^2$ .
Solution:
$(a^3b)^2 = a^{3 \times 2}b^2 = a^6b^2$
Answer: $a^6b^2$

Question 12

Problem: Simplify $\frac{\sqrt{48}}{\sqrt{147}}$ .
Solution:
$\frac{\sqrt{48}}{\sqrt{147}} = \sqrt{\frac{48}{147}} = \sqrt{\frac{16 \times 3}{49 \times 3}} = \sqrt{\frac{16}{49}} = \frac{\sqrt{16}}{\sqrt{49}} = \frac{4}{7}$
Answer: $\frac{4}{7}$

Question 13

Problem: Given $r = \frac{V}{I}$ , solve for V.
Solution:
$r = \frac{V}{I} \Rightarrow V = rI$
Answer: $V=Ir$

Question 14

Problem: Which system has no solution?
Analysis:
- a. y = x + 4, y - x = -4 => y = x + 4, y = x - 4. These lines are parallel and distinct, so no solution.
- b. 2y = 2x + 8, -2x = 2y - 8 => y = x + 4, -2x = 2y -8 => y = -x + 4. The second Equation can be re-arranged as y= -x + 4, has a solution. No Solution.
- c. y = 1/2 x + 6, 2x + 5 = y => y = 1/2 x + 6, 2x + 5 = y . Subsitute the second equation for y : 2x+ 5 = 1/2 x + 6 => (3/2) x= 1, has a solution.
- d. y = 4x + 1, y - 1 = 4x => y = 4x + 1, y = 4x + 1: Equations are the same, has a solution, infinitely many solutions.
Answer: a. y = x + 4, y - x = -4

Question 15

Problem: Venn diagram problem about people wearing blue or red shirts. Find the missing value.
Information: Total people = 30. From the diagram determine who wears only blue, only red, or both.
Analysis: Let x be the missing value (people who wear only red). Then, the sum of all sections of the Venn diagram should equal 30.
Answer: The missing value is 7.

Question 16

Problem: Distance between Kensington and Greenwich on a map.
Analysis: Based on the map (not provided), determine the scale and measure the distance.
Answer: 20 \sqrt{5} miles.

Question 17

Problem: Calculate the commission earned on a $4450 sale with a 3% commission rate.
Solution:
$Commission = 0.03 \times 4450 = 133.50$
Answer: $133.50

Question 18

Problem: Given $f(x) = x^2 + 1$ and domain D = {-2, -1, 0, 1, 3}, find the range R.
Solution:
- $f(-2) = (-2)^2 + 1 = 4 + 1 = 5$
- $f(-1) = (-1)^2 + 1 = 1 + 1 = 2$
- $f(0) = (0)^2 + 1 = 0 + 1 = 1$
- $f(1) = (1)^2 + 1 = 1 + 1 = 2$
- $f(3) = (3)^2 + 1 = 9 + 1 = 10$
Range: R = {5, 2, 1, 2, 10}

Question 19

Problem: Solve $y + w - \frac{3}{4}z = 0$ for z.
Solution:
$\frac{3}{4}z = y + w \Rightarrow z = \frac{4}{3}(y + w)$

Question 20

Problem: Gloria earns 1.5 times her normal pay for overtime (over 40 hours). She worked 47 hours and earned $489.85. Given $40p + 7(1.5p) = 489.85$ , find p.
Solution:
$40p + 10.5p = 489.85 \Rightarrow 50.5p = 489.85 \Rightarrow p = \frac{489.85}{50.5} = 9.70$
Answer: Gloria's normal hourly pay is $9.70.

Question 21

Problem: Determine if the line is positive, negative, zero or undefined
Answer: The slope is undefined.

Question 22

Problem: Given $A = {a, b, d, f, g}$ , $U = {\text{letters of the alphabet}}$ , and $A \cap B = {b, d}$ , which could be set B?
Analysis: B must contain b and d, and can contain other elements from U.
Answer: B = {b, d, k}

Question 23

Problem: Leah scored p points in the first half, and in the second half, she scored 3 more than 1/2 the points she scored in the first half. Altogether she scored 21 points. Given $p + (\frac{1}{2}p + 3) = 21$ , find p.
Solution:
$p + \frac{1}{2}p + 3 = 21 \Rightarrow \frac{3}{2}p = 18 \Rightarrow p = \frac{2}{3} \times 18 = 12$
Answer: Leah scored 12 points in the first half.

Question 24

Problem: Subtract $(6a^2 + 3a) - (4a^2 + 2a)$ .
Solution:
$6a^2 + 3a - 4a^2 - 2a = (6a^2 - 4a^2) + (3a - 2a) = 2a^2 + a$
Answer: $2a^2 + a$

Question 25

Problem: Find the equation of the line with x-intercept = -2 and y-intercept = -4.
Analysis: The line passes through (-2, 0) and (0, -4).
Slope: m = (-4 - 0) / (0 - (-2)) = -4 / 2 = -2
Equation: y = -2x - 4

Question 26

Problem: Janell has 5 gallons of paint. After painting 800 square feet, she has 3 gallons left. Find the equation of the linear function and interpret the slope.
Analysis: The points are (0, 5) and (800, 3) [ (sq ft, gallons)]
Slope: m = (3-5)/(800-0) = -2/800 = -1/400
Equation: y = (-1/400)x + 5
Interpretation: For every square foot painted, 1/400 th of a gallon is used
Answer: y = (-1/400)x + 5 slope = -1/400; this means that for every square foot painted, 1/400 gallons of paint is used.

Question 27

Problem: Which expression is NOT equivalent to the other expressions? a. (4x^2y)^2 c. 16x^4y^2 b. 4x^4y^2 d. 4^2x^4y^2
Solution: Let's check the options.
- (4x^2y)^2 = 16x^4y^2
- b. 4x^4y^2
- c. 16x^4y^2
- d. 4^2 x^4 y^2 = 16x^4y^2. Since expression b is different, that is answer.
Answer: b. 4x^4y^2

Question 28

Problem: The height of a ball is modeled by $y = -16x^2 + 72x$ , where x is the time in seconds. How long is the ball in the air?
Solution: The ball is in the air until y = 0.
$0 = -16x^2 + 72x = x(-16x + 72) =0$
x=0 , -16x + 72 = 0 => x = 72/16 = 9/2 = 4.5
Answer: 4.5 s

Question 29

Problem: Venn diagram for sets A and B. What is the intersection? Set A: factors of 9 Set B: factors of 12
Solution: A = {1, 3, 9}, B = {1, 2, 3, 4, 6, 12}
Intersection: $A \cap B = {1, 3}$
Answer: {1, 3}

Question 30

Problem: Factor the expression $p^2 - 40$
Since 40 is not a perfect square, we cannot factor the expression.
Answer: cannot be factored

Question 31

Problem: Multiply $(a+b)(a-b)$
Solution:
$(a+b)(a-b) = a^2 - ab + ab - b^2$
Answer: $a^2 - b^2$

Question 32

Problem: Simplify $y^{10} \cdot y^5$
Solution:
$y^{10} \cdot y^5 = y^{10+5}$
Answer: $y^{15}$

Question 33

Problem: Solve $7(x-2) = 7x +14$
Solution:
7(x-2) = 7x +14 => 7x - 14 = 7x + 14 => -14 = 14
Answer: No Solution

Question 34

Problem: Find the slope containing the points (1,-1) and (-2,8)
Solution:
$m = (y<em>2 - y</em>1)/(x<em>2 - x</em>1) = (8-(-1))/(-2-1)$
Answer: -3

Question 35

Problem: For f(x) = 24 - 2x, find f(2) and find x such that f(x)=10
Solution:
$f(2) = 24 - 2*2 = 20$ 10 = 24 - 2x=> -14 = -2x
Answer: 20; 7

Question 36

Problem: If you graph $y=x^2 -6x + 9$ the y-intercept of the graph of the equation is
Solution:
The y-intercept is the value of y when x= 0 substitute 0 for x: $y=0^2 -6(0) + 9$
Answer: 9

Question 37

Problem: Reserved tickets cost $20 each and General admission tickets cost $12 each. Total $900 sold. 20x+12y = 900.
Analysis:
Examine the graph and find the equation represented and the y intercept of graph
Answer: Need Graph to solve

Question 38

Problem: Give the domain and range of the relation
Analysis:
Based on the graph determine the domain (x values) and range (y values)
Answer: Need Graph to solve

Question 39

Problem: Solve $x^2 -7x - 8 = 0$ by factoring
Solution:
x^2 -7x - 8 = 0 => (x-8)(x+1) = 0 So we get $x=8 or x=-1$
Answer: $x=-1 or x=8$

Question 40

Problem: Which of the following graph shows the equation $y+1 = 2(x-1)$
Analysis:
Simplify the equation and analyze graph to find this equation.
Answer: Equation y+1 = 2(x-1) => y = 2x -3

Question 41

Problem: The scatter plot shows the relationship between the weekly total sales ($) and the number of different rug designs a rug store has. Based on this relationship, use the line of best fit to predict what the total sales will be when the store has 110 different rug designs.
Analysis:
Determine the liner of best fit to show the relationship between the two variables and use this line to predict sales amount
Answer: Need graph to solve this.

Question 42

Problem: Factor $x^2 - 16$
Solution:
Difference of the two squares: $x^2 - 16 = (x+4)(x-4)$
Answer: $(x+4)(x-4)$

Question 43

Problem: Factor $x^2 - 6x - 16$
Solution:
$x^2 - 6x - 16 = (x + 2)(x-8)$
Answer: $(x + 2)(x-8)$

Question 44

Problem: Solve $A = \frac{1}{2}(b+c)h$ for c
Solution:
2A = (b+c)h => 2A/h = b + c => c = 2A/h - b
Answer: $c = \frac{2A}{h} - b$

Question 45

Problem: The ratio of boys to girls in a class is 2:3. If there are 18 girls in the class, how many boys are there?
Solution:
\frac{boys}{girls} = \frac{2}{3} => boys = \frac{2}{3} * 18
Answer: 12

Question 46

Problem: Solve the system of equations
2x + 3y = 4 and 3x - 3y = -9
Solution: add the two equations to eliminate for y:
5x = -5 => so x = -1 now solve for y 2(-1) + 3y = 4.
Answer: (-1, 2)

Question 47

Problem: Use the zero product property to solve equation (x=3)(x-2) = 14
Solution:
Do not equate (x+3)=14 and (x-2) = 14
Rewrite the equation and multiply terms:(x+3)(x-2) = 14 = x^2 + x -6 = 14 set to 0: x^2+x -20 = 0
Answer: The solutions are -5, 4.

Question 48

Problem: Divide $\frac{18x^3 + 9x^2}{3x}$
Solution:
$\frac{18x^3 + 9x^2}{3x} = (18x^3)/(3x) + (9x^2)/(3x) = 6x^2 + 3x$
Answer: $6x^2 + 3x$

Question 49

Problem: Which of the following is the solution to this inequality? 3(5 + 2n) ≥ 7 + 10n
Solution:
15 + 6n ≥ 7 + 10n => 8 ≥ 4n => n ≤ 2
Answer: n ≤ 2

Question 50

Problem: Multiply (x+7)(x-7)
Solution:
$(x+7)(x-7) = x^2 -49$
Answer: x^2 - 49

Question 51

Problem: U is the set of natural numbers less than 8. G is the set of even integers less than 10. Which is the complement of set G in universe U?
Solution:
U = {1, 2, 3, 4, 5, 6, 7}, G = {2, 4, 6, 8}. Find all the values of U that are not in G.
Answer: {1, 3, 5, 7}

Question 52

Problem: Simplify the quotient $\frac{15}{\frac{2}{1}}$
Solution:
$\frac{15}{\frac{2}{1}} = \frac{15}{2}$
Answer: $\frac{15}{2}$

Question 53

Problem: Graph -2x + 4y = 4 for the domain D: {–8, –4, 0, 4, 8}.
Solution: Graph and evaluate for D values
Answer: requires the graph

Question 54

Problem: Determine whether the pairing is a function. If it is a function, describe the rule that relates the input value to the output value. input -3 -1 0 1 3 output 0 2 3 4 6
Answer: The pairing is: The pairing is a function. The rule is “input value plus 3.”

Question 55

Problem: The values in the table show a linear relationship. Find the slope. x −4 2 8 14 y 10 7 4 1
Solution: Find the slope
Answer: m= (7-10)/(2-(-4)) = (-3)/6 = -1/2