Solve the rational inequality. Write the solution set in interval notation. (WILL CHOOSE BEST ANSWER)?

(4x)/(-5x+18) ≥ 10

Since the rational expression is non-negative, the numerator and denominator must have the same sign.

Note -5x+18 ≠ 0 otherwise the expression is undefined.

a) If 4x ≥ 0 then -5x+18 > 0

4x ≥ 0 → x ≥ 0

-5x+18 > 0 → -5x > -18 → x < 18/5

4x ≥ -50x+180

54x ≥ 180

x ≥ 10/3

If 4x ≥ 0 then, 10/3 ≤ x < 18/5

b) If 4x ≤ 0 then -5x+18 < 0

4x ≤ 0 → x ≤ 0,

-5x+18 < 0 → -5x < -18 → x > 18/5

Note that this is an impossible condition because if x ≤ 0, then x cannot be greater than a positive number. Therefore the assumption that 4x ≤ 0 is a false assumption.

We are left with the condition that 4x ≥ 0 which leads to the conclusion that the interval containing x is:

[10/3, 18/5)

which is not shown in the list of choices.

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Note that D is not the correct choice since x can take on the value of 10/3 which is not contained in the interval (10/3, 18/5)

Let x = 10/3, then

4(x)/(-5x+18) ≥ 10

4(10/3)/(-5*10/3 + 18) ≥ 10

(40/3)/(-50/3 + 54/3) ≥ 10

40/(-50+54) ≥ 10

40/4 ≥ 10

10 ≥ 10 : which is a true statement showing that x = 10/3 is a valid solution of the inequality.