+5 Let a, b, c, and d be real numbers such that a > b and c > d. Enter the letters of the statements below that MUST be true.
a) a + c > b + d
b) 2a + 3c > 2b + 3d
c) a - c > b - d
d) ac > bd
e) a^2 + c^2 > b^2 + d^2
f) a^3 + c^3 > b^3 + d^3
g) a^100 + b^100 > c^100 + d^100
h) a + b > c + d
+1768 Let's analyze each statement based on the given conditions:
a) a + c > b + d: Since a > b and c > d, adding them together will preserve the inequality. So, a + c > b + d is MUST be true.
b) 2a + 3c > 2b + 3d: Multiplying both sides of a > b and c > d by positive constants 2 and 3, respectively, will still hold true. So, 2a + 3c > 2b + 3d MUST be true.
c) a - c > b - d: Subtracting c from both sides of a > b doesn't necessarily guarantee a - c > b - d. It depends on the relative values of a, b, and c. Not necessarily true.
d) ac > bd: Since a > b and c > d, multiplying them together will maintain the inequality as long as both a and c have the same sign (both positive or both negative). This is MUST be true.
e) a^2 + c^2 > b^2 + d^2: Squaring an inequality doesn't necessarily preserve the direction of the inequality. It depends on the signs of a and b. Not necessarily true.
f) a^3 + c^3 > b^3 + d^3: Similar to case (e), cubing an inequality doesn't guarantee the same direction for the result. Not necessarily true.
g) a^100 + b^100 > c^100 + d^100: The behavior of high-power exponents is unpredictable with inequalities. We cannot determine the direction of the inequality based on the given information. Not necessarily true.
h) a + b > c + d: Since a > b and c > d, adding b to both sides of a > b doesn't necessarily make the sum larger than c + d. It depends on the relative values of a, b, c, and d. Not necessarily true.
Summary: The statements that MUST be true based on the given conditions are:
a) a + c > b + d
b) 2a + 3c > 2b + 3d
d) ac > bd
Therefore, the answer is a, b, and d.
