For positive integers m and p, x
Answer(s): A
Write x = d
Frequency distributions A and B each show the distribution of 47 integers.Quantity AThe median of the 47 integers in frequency distribution AQuantity BThe median of the 47 integers in frequency distribution B
Answer(s): C
Both histograms show 47 integers: 3 + 7 + 9 + 12 + 7 + 6 + 3 = 47 for A and likewise 3 + 6 + 9 + 12 + 8 + 6 + 3 = 47 for B.Because 47 is odd, the median is the 24th number when the data are listed in order.Distribution ACumulative count up to each value:1 3, 2 10, 3 19, 4 31.The 24th number is included in the group of 12 fours, so median A = 4.Distribution BCumulative count:1 3, 2 9, 3 18, 4 30.Again, the 24th number falls in the group of 12 fours, so median B = 4.Hence both medians are equal.
a < 0b < 0Quantity AIn the xy-plane, the slope of the line containing the points (a, b) and (0, 2)Quantity BIn the xy-plane, the slope of the line containing the points (a, b) and (2, 0)
Let the two slopes be mA = slope of the line through (a, b) and (0, 2)mB = slope of the line through (a, b) and (2, 0)Write each slope as a fraction:Note: denominator a is negative, denominator (2 - a) is positive because a < 0.Compare mA and mB:Multiply both sides by the product a(2 - a). Because a < 0 and (2 - a) > 0, their product is negative. Multiplying by a negative quantity reverses the inequality:(b - 2) (2 - a) < -ba(b - 2) (2 - a) = 2b - 4 - ab + 2a2b - 4 + 2a < 02(a + b - 2) < 0Since both a and b are negative, a + b - 2 is certainly negative, making the inequality true.Therefore, mA is greater than mB.
x > 5Quantity AQuantity B
Since x > 5, both denominators are positive, so inequality direction is preserved.Let's test x = 6:Try x = 10:Try x = 100:Since x is always greater than 5, and x is positive,
The measure of one angle of a triangle is 30 degrees. The two line segments forming the angle have lengths x and 5. The length of the third side of the triangle is 3.Quantity AxQuantity B4
Answer(s): D
For any triangle with angle between sides a and b, and opposite side c, the Law of Cosines says:c2 = a2 + b2 - 2ab × cos()Here:a = x b = 5c = 3= 30°Substitute:Solve the quadratic:Both values satisfy the triangle inequality with sides 5 and 3, so each can occur in an actual triangle. Because x can be less than 4 in one valid case and greater than 4 in another, no single inequality between x and 4 is guaranteed.
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