Algebra I: Master Linear Equations, Inequalities, and Functions

Question 1

Which graph represents the equation y = 2x + 1?

Accepted Answer

A line with a slope of 2 and a y-intercept of 1

Answer

A line with a slope of 1 and a y-intercept of 2

Answer

A parabola with a vertex at (1, 2)

Question 2

Solve the inequality -2x + 5 < 11. Explain why the inequality symbol changes when dividing both sides by -2.

Accepted Answer

x < 3. Dividing by -2 reverses the inequality because it changes the direction of the inequality.

Answer

x > 3. Multiplying by -2 reverses the inequality.

Answer

x < -3. Dividing by -2 keeps the inequality the same because -2 is positive.

Answer

x > -3. Dividing by -2 keeps the inequality the same because -2 is negative.

Question 3

Graph the function: y = -x + 2

Accepted Answer

A straight line with a slope of -1 and a y-intercept of 2

Answer

A straight line with a slope of 1 and a y-intercept of 2

Answer

A curved line with a slope of 1 and a y-intercept of -2

Answer

A curved line with a slope of -1 and a y-intercept of -2

Question 4

Simplify the expression: (x - 2)(x + 3)

Accepted Answer

x^2 + x - 6

Answer

x^2 - x - 6

Answer

x^2 + x + 6

Answer

x^2 - x + 6

Question 5

Which inequality represents 'The number x is greater than 5'?

Accepted Answer

x > 5

Answer

x >= 5

Answer

x < 5

Answer

x <= 5

Question 6

Find the slope of the line passing through the points (1, 3) and (4, 7)

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 7

Factor the polynomial: x^2 - 4

Accepted Answer

(x - 2)(x + 2)

Answer

(x - 4)(x + 1)

Answer

(x - 2)

Answer

(x + 4)

Question 8

Determine the domain and range of the linear function f(x) = 2x + 1. Provide a clear explanation of your reasoning.

Accepted Answer

Domain: All real numbers, Range: All real numbers

Answer

Domain: All positive numbers, Range: All positive numbers

Answer

Domain: All negative numbers, Range: All negative numbers

Answer

Domain: All real numbers except 0, Range: All real numbers except 0

Question 9

Find the solution to the system of linear equations: 
x + y = 5
x - y = 1

Accepted Answer

x = 3, y = 2

Answer

x = 2, y = 3

Answer

x = 3, y = -2

Answer

x = -2, y = 3

Question 10

Which of the following functions is linear?

Accepted Answer

y = 2x + 5

Answer

y = x^2

Answer

y = sin(x)

Answer

y = e^x

Question 11

Find the inverse of the function f(x) = 2x - 1.

Accepted Answer

f^-1(x) = (x + 1)/2

Answer

f(x) = 2x + 1

Answer

f^-1(x) = 1/2x - 1

Question 12

Which system of inequalities defines the solution region of the following system of linear inequalities: x + y < 6 and x - y > -2?

Accepted Answer

x + y < 6 and x - y > -2

Answer

x < 6 and y < 2

Answer

x + y > 6 and x - y < -2

Answer

x - y < 6 and x + y > -2

Question 13

Determine if the function f(x) = x^2 + 2x - 3 is a linear function.

Accepted Answer

No

Answer

Yes

Question 14

Determine the slope and y-intercept of the linear function: f(x) = -3x + 5

Accepted Answer

Slope: -3, Y-intercept: 5

Answer

Slope: 3, Y-intercept: -5

Answer

Slope: -5, Y-intercept: 3

Answer

Slope: 5, Y-intercept: -3

Question 15

Consider the statement: "For any two linear functions, their graphs must intersect at exactly one point." Explain why this statement is false and provide a specific example to support your answer.

Accepted Answer

The statement is false. Two parallel lines never intersect.

Answer

The statement is true for all linear functions.

Answer

The statement is true only for non-parallel lines.

Answer

The statement is true only for parallel lines.

Question 16

Determine the equation of the line with a slope of 3 that passes through the point (0, -2).

Accepted Answer

y = 3x - 2

Answer

y = 2x - 3

Answer

y = -3x - 2

Answer

y = 3x + 2

Question 17

Which graph represents the equation y = |x - 1|?

Accepted Answer

Graph C: A V-shaped graph

Answer

Graph A: A straight line with a slope of 2 and a y-intercept of 1

Answer

Graph B: A parabola opening upwards

Question 18

Which function has a domain of all real numbers and a range of [0, ∞)?

Accepted Answer

y = |x|

Answer

y = x^2

Answer

y = 1/x

Question 19

Which of the following transformations would result in the graph of y = 2x + 3 shifting 3 units to the left?

Accepted Answer

y = 2(x - 3) + 3

Answer

y = 2(x + 3) + 3

Answer

y = 2x - 3

Answer

y = 2x + 6

Question 20

Which region represents the solution to the linear inequality 2x - 3y > 6?

Accepted Answer

The area above the line 2x - 3y = 6

Answer

The area below the line 2x - 3y = 6

Answer

The entire coordinate plane

Answer

There is no solution region

Question 21

Which graph accurately represents the quadratic function y = -x^2 + 2x + 1?

Accepted Answer

A. [Image of a parabola opening downward, with vertex at (1, 2)]

Answer

B. [Image of a parabola opening upward, with vertex at (1, 2)]

Answer

C. [Image of a straight line with a positive slope]

Answer

D. [Image of a circle]

Question 22

When graphing the linear equations y = 2x + 1 and y = -x + 3 on the same coordinate plane, how do their slopes and intercepts affect the steepness and position of the lines?

Accepted Answer

The slope determines the steepness, and the intercept determines the starting point on the y-axis.

Answer

The intercept determines the steepness, and the slope determines the starting point on the y-axis.

Answer

Both the slope and intercept determine the starting point on the y-axis.

Answer

Neither the slope nor the intercept affects the steepness or position of a line.

Question 23

Determine the equation of the line that is perpendicular to y = 2x - 1 and intersects the point (1, 3).

Accepted Answer

y = -1/2x + 4

Answer

y = 2x + 1

Answer

y = 1/2x + 2

Question 24

Two lines have different y-intercepts but the same slope. Which statement must be true?

Accepted Answer

The lines are parallel.

Answer

The lines are perpendicular.

Answer

The lines intersect.

Answer

The lines never intersect.

Question 25

What are the vertex, domain, and range of the function f(x) = |x - 2| + 1?

Accepted Answer

Vertex: (2, 1), Domain: All real numbers, Range: y ≥ 1

Answer

Vertex: (0, 1), Domain: x ≥ 0, Range: y ≥ 1

Answer

Vertex: (0, 1), Domain: All real numbers, Range: y ≥ 0

Question 26

Which inequality represents the solution set of x - 5 > 7?

Accepted Answer

x > 2

Answer

x < 2

Answer

x < 12

Answer

x > 12

Question 27

Find the solution to the system of equations: y = x + 2 and 2x - y = 4

Accepted Answer

x = 2, y = 4

Answer

x = 3, y = 5

Answer

x = 4, y = 6

Answer

x = 5, y = 7

Question 28

What is the domain of the expression √(2x - 1)?

Accepted Answer

{x | x ≥ 1/2}

Answer

{x | x ≤ 1/2}

Answer

{x | x > 1/2}

Answer

{x | x < 1/2}

Question 29

Solve the system of equations:

2x + 3y = 6
4x + 6y = 12

How many solutions does this system have?

Accepted Answer

Infinitely many

Answer

No solutions

Answer

Exactly one solution

Question 30

Analyze the graph of the exponential function f(x) = 2 ^ x. Determine its domain, range, and vertical asymptote.

Accepted Answer

Domain: (-∞, ∞); Range: (0, ∞); Vertical Asymptote: x = 0

Answer

Domain: (0, ∞); Range: (-∞, ∞); Horizontal Asymptote: y = 0

Answer

Domain: (-∞, 0); Range: (0, 2); Oblique Asymptote: y = 2x

Answer

Domain: (0, ∞); Range: (∞, -∞); Vertical Asymptote: x = 2

Question 31

A population of bacteria doubles every hour. If there are initially 100 bacteria, write an equation to model the population growth, and how many bacteria will there be after 6 hours?

Accepted Answer

P(t) = 100 * 2^t

Answer

P(t) = 100 * t^2

Answer

P(t) = 100 * 3^t

Answer

P(t) = 100 * e^t

Question 32

Find the value of x that makes the expression undefined:
1/(x - 2)

Accepted Answer

2

Answer

3

Answer

0

Answer

-1

Question 33

Which of the following is NOT a quadratic equation?

Accepted Answer

y = 2x + 3

Answer

y = 3x^2 - 5x + 2

Answer

y = x^2 + 2x + 1

Answer

y = -x^2 + 4x - 3

Question 34

Factor the expression:
x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 3)(x + 3)

Answer

(x - 3)(x - 3)

Answer

(x + 9)(x - 1)

Question 35

Consider the linear equation **y = -2x + 4**. Which of the following graphs accurately represents this equation?

Accepted Answer

A graph with a y-intercept at 4 and a slope of -2

Answer

A graph with a y-intercept at 0 and a slope of 2

Answer

A graph with a y-intercept at -4 and a slope of 2

Question 36

Solve the equation: **|x - 5| = 3**.

Accepted Answer

x = 2  or x = 8

Answer

x = 3

Answer

x = 5

Answer

x = 10

Question 37

Simplify the radical expression: **√(12)**.

Accepted Answer

2√3

Answer

6

Answer

4

Answer

3

Question 38

Which expression is equivalent to **(x + y)^2**?

Accepted Answer

x^2 + 2xy + y^2

Answer

x^2 - y^2

Answer

2x^2 + 2xy + 2y^2

Answer

x^2 - 2xy + y^2

Question 39

Simplify the algebraic expression: (3x + 2)(x - 1)

Accepted Answer

3x² - x - 2

Answer

3x² - x + 2

Answer

3x² + x + 2

Answer

3x² + x - 2

Question 40

Find the slope of the line passing through the points (2, 5) and (4, 9)

Accepted Answer

2

Answer

1

Answer

-1

Answer

-2

Question 41

Factor the expression x^2 - 9

Accepted Answer

(x + 3)(x - 3)

Answer

(x + 9)(x - 9)

Answer

(x + 3)(x + 3)

Answer

(x - 3)(x - 3)

Question 42

Graph the linear function y = -2x + 1.

Accepted Answer

Image of a line with a slope of -2 and a y-intercept of 1

Answer

Image of a parabola opening upwards

Answer

Image of a line with a slope of 2 and a y-intercept of -1

Question 43

Simplify the algebraic expression: (2x^2) * (3x)

Accepted Answer

6x^3

Answer

2x^3

Answer

6x^2

Answer

x^5

Question 44

Simplify the algebraic expression: (x + 3)^2

Accepted Answer

x^2 + 6x + 9

Answer

x^2 - 9

Answer

x^2 - 6x + 9

Answer

x^2 + 9

Question 45

Solve the radical equation: √(x + 5) = 3

Accepted Answer

x = 4

Answer

x = -4

Answer

x = -9

Answer

x = 9

Question 46

Which of the following equations represents a line with a slope of 3 and a y-intercept of -2?

Accepted Answer

y = 3x - 2

Answer

y = -3x + 2

Answer

y = -2x - 3

Answer

y = 2x + 3

Question 47

Graph the inequality: x < 2

Accepted Answer

A shaded region to the left of the vertical line x = 2

Answer

A closed circle at x = 2

Answer

A shaded region to the right of the vertical line x = 2

Question 48

Simplify the expression: (2x^2y)^3

Accepted Answer

8x^6y^3

Answer

4x^6y^3

Answer

6x^6y^3

Question 49

Evaluate the expression: log base 3 of 27

Accepted Answer

3

Answer

12

Answer

9

Answer

6

Question 50

Which of the following is the graph of the linear equation y equals 2x minus 1?

Accepted Answer

A straight line with a y-intercept of -1 and a slope of 2

Answer

A circle with a radius of 1

Answer

A parabola with a vertex at (0, -1)

Answer

A straight line with a y-intercept of 1 and a slope of 2

Question 51

What is the value of *x* in the equation 2*x* + 5 = 13?

Accepted Answer

4

Answer

5

Answer

3

Answer

6

Question 52

Which number is *not* a solution to the inequality *x* - 3 > 5?

Accepted Answer

10

Answer

8

Answer

9

Answer

7

Question 53

What is the slope of the line that passes through the points (2, 5) and (4, 9)?

Accepted Answer

2

Answer

3

Answer

4

Answer

1

Question 54

What is the equation of the line that has a slope of -2 and passes through the point (1, 3)?

Accepted Answer

y = -2x + 5

Answer

y = -2x + 3

Answer

y = 2x + 3

Answer

y = 2x - 1

Question 55

What are the values of *x* and *y* that solve the system of equations: *x* + *y* = 5 and *x* - *y* = 1?

Accepted Answer

x = 3, y = 2

Answer

x = 4, y = 1

Answer

x = 2, y = 3

Answer

x = 1, y = 4

Question 56

Which of the following is an example of a polynomial expression?

Accepted Answer

x^2 + 2x - 3

Answer

√x

Answer

1/x

Answer

2x + 5

Question 57

Simplify the expression: (2x^2y) / (xy^2)

Accepted Answer

2x/y

Answer

2y/x

Answer

y/2x

Answer

x/2y

Question 58

Factor the expression: x^2 - 4

Accepted Answer

(x + 2)(x - 2)

Answer

(x - 2)^2

Answer

(x + 2)^2

Answer

(x + 4)(x - 4)

Question 59

What is the value of *x* in the equation √(x + 5) = 7?

Accepted Answer

44

Answer

49

Answer

25

Answer

12

Question 60

What is the domain of the function f(x) = 1/(x - 2)?

Accepted Answer

All real numbers except 2

Answer

All real numbers

Answer

All negative real numbers

Answer

All positive real numbers