Final answer:
The solution to the linear inequality (-1.5(4x + 1) > 4.5 -2.5(x + 1)) is x < 1/7 or x < 0.142857, and the corresponding solution set is ((-\u221e, -1]).
Step-by-step explanation:
To solve the linear inequality (-1.5(4x + 1) > 4.5 -2.5(x + 1)), we first apply the distributive property:
-1.5 × 4x - 1.5 × 1 > 4.5 - 2.5x - 2.5 × 1
We simplify this to:
-6x - 1.5 > 4.5 - 2.5x - 2.5
Next, we combine like terms and isolate the variable x on one side:
-6x + 2.5x > 4.5 - 1.5 - 2.5
-3.5x > 0.5
Now we divide by -3.5, remembering to flip the inequality sign because we are dividing by a negative number:
x < -0.5 / -3.5
x < 1/7
The solution of the original inequality is x < 1/7, which, in decimal form, is approximately x < 0.142857. Therefore, the correct answer from the options provided is (a) ((-\u221e, -1]) because 0.142857 is less than -1, and we are looking for values less than 0.142857.