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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 first determine g using the given average. The average of g and 100 is calculated as (g + 100) / 2, which equals 75. By solving for g, we can then easily calculate g + 100.
A) 50
This choice suggests that g + 100 equals 50, which would imply that g is -50. However, this contradicts the given average of 75, as substituting g = -50 into the average formula yields a negative result, which is inconsistent with the arithmetic mean calculated from positive numbers.
B) 125
If g + 100 were 125, it would mean that g is 25. Substituting g = 25 into the average formula results in (25 + 100) / 2 = 62.5, which does not match the stated average of 75. Thus, this choice is incorrect.
C) 150
This is the correct choice. If g + 100 equals 150, then g must be 50. Substituting g = 50 into the average formula gives (50 + 100) / 2 = 75, which confirms that this option satisfies the condition provided in the question.
D) 175
If g + 100 were 175, it would imply g is 75. Calculating the average with g = 75 results in (75 + 100) / 2 = 87.5, which is not equal to the specified average of 75. Therefore, this option is also incorrect.
Conclusion
To solve for g + 100, we start with the average equation (g + 100) / 2 = 75 and find that g must equal 50. This leads directly to the conclusion that g + 100 equals 150, confirming that choice C is the accurate answer, while all other options fail to satisfy the initial average condition.
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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
This inequality describes a region in the xy-plane where the value of y is bounded above by the line y = 3x and below by the line y = -x. The graphical representation shows that y must satisfy both conditions simultaneously, creating a triangular region between these two lines.
A) x ≤ y ≤ 3x
This inequality suggests that y is greater than or equal to x and less than or equal to 3x. The line y = x would be the lower boundary, which does not match the graphical representation, as the lower boundary is actually y = -x. Therefore, this choice is incorrect as it misrepresents the lower limit of y.
B) x ≥ y ≥ 3x
This option indicates that y is less than or equal to x and greater than or equal to 3x. This scenario is impossible because y cannot be both less than or equal to x and greater than or equal to 3x simultaneously. The concept defines an invalid region, making this choice incorrect.
D) –x ≥ y ≥ 3x
This inequality implies that y is less than or equal to –x and greater than or equal to 3x. Similar to option B, this combination is also impossible; y cannot simultaneously be below –x and above 3x. Thus, this choice does not reflect the region depicted in the graph.
Conclusion
The correct inequality, –x ≤ y ≤ 3x, accurately captures the graphical representation in the xy-plane, delineating a region where y fluctuates between the lines y = -x and y = 3x. The incorrect options either misinterpret the boundaries or present logically impossible scenarios, underscoring the importance of understanding inequalities in graphical contexts.
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Question 3
In the inequalities, a and b are integers. What is the least integer value of a/b​? −5≤a≤5 1<b<5
Your Answer: Option(s)
Correct Answer: Option(s) C
Rationale
-2
To determine the least integer value of ( \frac{a}{b} ) given the inequalities ( -5 leq a leq 5 ) and ( 1 < b < 5 ), we must examine the possible values for (a) and ( b ). The integer ( b ) can take values of 2, 3, or 4, and we need to find the minimum ratio ( \frac{a}{b} ).
A) 2
The integer value of 2 can be achieved by taking ( a = 4 ) and ( b = 2 ), resulting in ( \frac{4}{2} = 2 ). However, this is not the least integer value since there are combinations that yield a smaller ratio.
B) -1
To achieve a ratio of -1, one would need ( a = -2 ) and ( b = 2 ), resulting in ( \frac{-2}{2} = -1 ). While this is a valid option, it is not the least integer value obtainable from the given ranges.
C) -2
Choosing ( a = -4 ) and ( b = 2 ) gives ( \frac{-4}{2} = -2 ), which is indeed the least integer value for ( \frac{a}{b} ) within the specified constraints. No other combination provides a smaller value than -2.
D) -4
For ( \frac{a}{b} = -4 ), one would need ( a = -4 ) and ( b = 1 ). However, since ( b ) must be greater than 1, this combination is invalid. Thus, -4 cannot be achieved under the given constraints.
Conclusion
The least integer value of ( \frac{a}{b} ) determined from the inequalities ( -5 leq a leq 5 ) and ( 1 < b < 5 ) is -2. This is found by selecting appropriate integer values for (a) and ( b ) that satisfy all conditions, demonstrating that while other options exist, -2 remains the minimum achievable ratio.
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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
A) 1755845659cA.png
To satisfy the inequality -1 <= 4x - 5, we can rearrange it to find the range of values for x. Solving the inequality gives us x ≥ 1, which corresponds to the correct graphical representation showing all values greater than or equal to 1.
B) 1755845713cB.png
This option likely represents a range of values that do not include x = 1 or greater, which does not satisfy the inequality. The graphical representation must show values that include all numbers starting from 1 and extending to infinity to be correct.
C) 1755845777cC.png
This choice probably depicts values that either start below 1 or do not encompass the entire range required by the inequality. An appropriate solution would need to clearly include all values from 1 onward, which this option fails to show.
D) 1755845819cD.png
This option likely represents values that either cap off before reaching 1 or incorrectly includes values less than 1. The inequality clearly specifies that x must be greater than or equal to 1, which this representation does not meet.
Conclusion
In solving the inequality -1 <= 4x - 5, we find that x must be greater than or equal to 1. The correct representation of this solution is shown in option A, which includes all values of x from 1 to infinity. The other options fail to encompass this critical range, making them incorrect for the given inequality.
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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 of the line indicates a slope of 31 and a y-intercept of -1, which matches the visual representation on the graph, confirming that this is the correct choice among the options provided.
A) y=−31x+1
This equation features a slope of -31, which implies that the line would descend steeply from left to right. However, the line graphed rises steeply instead, indicating a positive slope, thus making this option incorrect.
B) y=31x−1
This equation accurately depicts a line with a steep positive slope of 31 and a y-intercept at -1. Both characteristics correspond perfectly with the visual representation of the line on the graph, confirming its correctness.
C) y=3x-1
While this equation has the correct y-intercept of -1, the slope of 3 is too shallow compared to the steepness of the graphed line. The line represented by this equation would not match the steep incline visible in the graph, rendering this option incorrect.
D) y=−3x−9
This equation has a negative slope of -3, indicating a downward trend from left to right, which contradicts the positive trend of the line depicted in the graph. Additionally, the y-intercept of -9 is far below the actual intercept observed, making this option incorrect.
Conclusion
Among the options provided, only y=31x−1 accurately represents the equation of the line shown in the graph. It possesses both the correct positive slope and y-intercept, aligning perfectly with the line's trajectory. The other choices either misrepresent the slope or y-intercept, thereby confirming that they do not correspond to the graphed line's properties.
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