Final answer:
To sort the expressions given that x is negative, we simplify where possible and compare the positive constants. We do not change the relative order by multiplying by x, so we sort the constants 6√3, 12.25, 9.8, and 11.4, and list 57x/5, 6x√3 (or x√75), 9.8x, and x(3.5)² accordingly from greatest to least.
Step-by-step explanation:
The question involves ordering the given expressions containing a variable x when x is a negative real number. The expressions are: 6x√3, x(3.5)², x√75, 9.8x, and 57x/5. Since x is negative, multiplying by x will not change the relative order since multiplication rules for the sign tell us that the product of two negative numbers is positive.
However, we need to compare the positive constants to list these expressions from greatest to least. We can simplify some expressions further: (3.5)² becomes 12.25, and √75 simplifies to 5√3. Once simplified, we can clearly see 6x√3 and x√75 are equivalent. We essentially sort the constants 12.25, 9.8, and 11.4 (from 57/5) along with 6√3 to determine the order when multiplied by a negative x.
To sort the expressions from greatest to least considering a negative x value: 57x/5, 6x√3 (or x√75), 9.8x, x(3.5)².