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Question 1
If the average (arithmetic mean) of g and 100 is 75, what is the value of g + 100?
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
g + 100 equals 150.
To find the value of g + 100, we start with the average of g and 100 being 75. By solving the equation for g, we determine that g + 100 must equal 150.
A) 50
If g + 100 were 50, this would imply that g equals -50. Plugging this back into the average calculation gives an average much less than 75, as the average would be (-50 + 100) / 2 = 25, which does not satisfy the initial condition.
B) 125
If g + 100 were 125, then g would equal 25. The average of 25 and 100 would be (25 + 100) / 2 = 62.5, which is also less than 75, thus failing to meet the requirement set by the problem.
C) 150
If g + 100 equals 150, then g would be 50. The average of 50 and 100 is (50 + 100) / 2 = 75, which perfectly aligns with the condition given in the question.
D) 175
If g + 100 were 175, this would imply that g equals 75. The average of 75 and 100 would be (75 + 100) / 2 = 87.5, which exceeds 75, thereby contradicting the problem's stipulation.
Conclusion
The correct value of g + 100 is 150, derived from the condition that the average of g and 100 is 75. By isolating g, we confirm that only choice C satisfies the average condition, while the other options lead to values of g that do not align with the specified average. Thus, the arithmetic mean condition uniquely identifies the correct answer.
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Question 2
Which of the following inequalities is graphed in the xy-plane above?
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
−x≤y≤3x
The inequality −x ≤ y ≤ 3x indicates that y is bounded above by the line y = 3x and below by the line y = -x. This means that for any given x-value, y can take on values between these two lines, which shapes the region represented in the xy-plane.
A) x≤y≤3x
This choice states that y is bounded below by the line y = x and above by the line y = 3x. The graph of this inequality would depict a region that starts from the line y = x, which is incorrect since the provided graph has y starting from y = -x instead. Thus, this choice does not accurately represent the graphed region.
B) x≥y≥3x
This option suggests that y is bounded above by x and below by 3x. However, this interpretation implies that y is less than or equal to x, which is not consistent with the graph showing a region above the line y = -x. Hence, this choice is also incorrect.
C) −x≤y≤3x
This is the correct choice, as it accurately captures the region between the lines y = -x and y = 3x. The graph illustrates that for any value of x, y can indeed take values between these two lines, matching the conditions set by the inequality.
D) −x≥y≥3x
This choice indicates that y is bounded above by -x and below by 3x. This scenario does not match the provided graph, as it would represent a region below the line y = -x rather than above it. Therefore, this option is incorrect.
Conclusion
The inequality −x ≤ y ≤ 3x correctly describes the region between the lines y = -x and y = 3x in the xy-plane, allowing y to vary within these bounds. The other options either misrepresent the relationships between x and y or depict regions that do not align with the graph provided, confirming that choice C is the only accurate representation.
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Question 3
In the inequalities, a and b are integers. What is the least integer value of a/bâ€â€¹? −5≤a≤5 1<b<5
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
-2
To find the least integer value of \( \frac{a}{b} \) given the constraints \(-5 \leq a \leq 5\) and \(1 < b < 5\), we need to consider the possible integer values of \(b\) and the corresponding values of \(a\). The minimum ratio occurs when \(a\) is at its lowest value and \(b\) is at its highest value.
A) 2
For \( \frac{a}{b} \) to equal 2, \(a\) would need to be at least 2 times \(b\). The highest possible value of \(b\) is 4 (since \(1 < b < 5\)), giving \(a\) a minimum value of 8, which is not allowed since \(a\) must be between -5 and 5. Thus, this choice is not feasible.
B) -1
If \( \frac{a}{b} = -1\), then \(a\) would need to equal \(-b\). The maximum \(b\) can be is 4, which would yield \(a = -4\). Although \(-4\) is within the bounds for \(a\), it is not the least value achievable; therefore, this option is incorrect.
C) -2
Setting \( \frac{a}{b} = -2\) requires \(a = -2b\). For \(b = 4\), \(a\) would be \(-8\), which is out of bounds. However, for \(b = 3\), \(a\) would equal \(-6\), still invalid. But for \(b = 2\), \(a = -4\) is valid, leading to the least integer value of \(-2\) for \( \frac{a}{b} \), making this the correct choice.
D) -4
If \( \frac{a}{b} = -4\), then \(a\) would need to be \(-4b\). The maximum permissible value for \(b\) is 4, which would give \(a = -16\), clearly outside the specified range for \(a\). Hence, this choice is not possible.
Conclusion
The problem requires finding the least integer value of \( \frac{a}{b} \) under the given constraints. After evaluating all options, it is evident that \(-2\) is the least achievable ratio when \(a = -4\) and \(b = 2\). Other choices either exceed the limits for \(a\) or do not yield a lesser value, confirming that \(-2\) is indeed the correct answer.
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Question 4
Which of the following represents all values of x that satisfy the inequality - 1 <= 4x - 5?
Your Answer: Option(s)
Correct Answer: Option(s) A
Rationale
-1 <= 4x - 5 is satisfied by the values shown in option A.
To solve the inequality -1 <= 4x - 5, we first isolate x. Adding 5 to both sides gives us 4 <= 4x, and dividing by 4 yields x >= 1. Thus, the solution includes all values of x that are greater than or equal to 1, which corresponds to the representation in option A.
A) 1755845659cA.png
This option accurately depicts all values of x that satisfy the inequality, which is x >= 1. It visually represents the solution on a number line, showing that all values from 1 to positive infinity are included.
B) 1755845713cB.png
This option incorrectly suggests that the solution starts at a number greater than 1. It might indicate that values less than 1 are included, which contradicts the derived solution of x >= 1. Therefore, this option does not satisfy the inequality.
C) 1755845777cC.png
This choice fails to represent the solution as it may only show a single point or values that are not inclusive of 1. The solution requires all x values from 1 and upwards, and this option does not reflect that.
D) 1755845819cD.png
Similar to option C, this choice likely illustrates values that do not include the necessary range starting from 1. The representation does not align with the solution of x >= 1, thus making it incorrect.
Conclusion
The correct representation of the solution to the inequality -1 <= 4x - 5 is option A, which correctly indicates that x must be greater than or equal to 1. The other options fail to satisfy this condition, highlighting the importance of accurate representation in mathematical inequalities.
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Question 5
Which of the following is an equation of the line graphed above?
Your Answer: Option(s)
Correct Answer: Option(s) B
Rationale
y=31x−1 represents the equation of the line graphed above.
The equation indicates a positive slope of 31 and a y-intercept of -1, consistent with the characteristics of the line shown in the graph.
A) y=−31x+1
This equation suggests a negative slope of -31, which would indicate a steep descent from left to right. However, the graph shows an upward trend, meaning the slope must be positive, disqualifying this option.
B) y=31x−1
This equation correctly describes the line's upward slope of 31 and a y-intercept of -1. It aligns perfectly with the graph's characteristics, confirming that this is the correct representation of the line.
C) y=3x-1
While this equation has the same y-intercept of -1, it features a much gentler slope of 3. The graph depicts a much steeper incline, indicating this option does not accurately represent the line's steepness.
D) y=-3x-9
This equation indicates a negative slope of -3 and a y-intercept of -9, suggesting the line would descend sharply from left to right and start much lower on the y-axis. This contradicts the graph's upward trajectory and its position on the coordinate plane.
Conclusion
The equation of the line graphed above must possess a steep positive slope and a specific y-intercept, which is best encapsulated by y=31x−1. Other options either feature incorrect slopes or y-intercepts that misrepresent the line’s behavior as depicted in the graph. Understanding these attributes is essential for accurately interpreting linear equations in the context of graphical representations.
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