Math

QuestionWhich of these expressions is always negative for a<0a<0 and b>0b>0? A. a+ba+b B. bab-a C. aba-b D. 2(ab)-2(a b) E. ba-\frac{b}{a}

Studdy Solution

STEP 1

Assumptions1. aa is a negative number. bb is a positive number

STEP 2

We need to determine which of the given expressions will always be negative given the conditions. Let's start by examining each option individually.
Option A a+ba+b

STEP 3

Since aa is negative and bb is positive, the sum a+ba+b could be positive, negative, or zero, depending on the absolute values of aa and bb. Therefore, this expression is not always negative.
Option B bab-a

STEP 4

Since aa is negative, a-a is positive. Therefore, bab - a is the sum of two positive numbers, which is always positive. So, this expression is not always negative.
Option C aba-b

STEP 5

Since aa is negative and bb is positive, aba - b is the difference of a negative number and a positive number, which is always negative. So, this expression is always negative.
Option D 2(ab)-2(ab)

STEP 6

Since aa is negative and bb is positive, the product abab is negative. Multiplying a negative number by 2-2 gives a positive number. So, this expression is not always negative.
Option ba-\frac{b}{a}

STEP 7

Since aa is negative and bb is positive, the fraction ba\frac{b}{a} is negative. Multiplying a negative number by 1-1 gives a positive number. So, this expression is not always negative.

STEP 8

After examining all the options, we find that only option C aba - b is always negative given the conditions.
So, the solution to the problem is option C aba - b.

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