Problem Solving

BTGmoderatorDC wrote:If a < b < 0, which of the following must be true?

A. a^2 < b^2
B. b − 10 < a
C. b + a > a
D. ab < b^2
E. ab < a^2

OA E

Source: Veritas Prep

Given that a < b < 0, we know that both a and b are negative and the absolute value of a is greater than that of b; thus, though a < b, we have |a| > |b|.

We can take suitable values for a and b. Say a = - 3 and b = -2.

Lets' take each option one by one.

A. a^2 < b^2

=> (-3)^2 < (-2)^2 => 9 < 4. This is incorrect since 9 > 4.

B. b - 10 < a

=> -2 - 10 < -3 => -12 < -3. This is correct; however is this must be correct?

Let's take another extremem example; say b = -2 and a = -15. Thus, from b - 10 < a, we have -2 - 10 < -15 => -12 < -15. This is incorrect since -15 < -12.

C. b + a > a

Say a = - 3 and b = -2. Thus, from b + a > a => b > 0; cancelling a from both sides. This is correct since b < 0.

D. ab < b^2

Say a = - 3 and b = -2. Thus, from ab < b^2 => -2 *- 3 < (-2)^2 => 6 < 4. This is correct since 6 > 4.

E. ab < a^2

Option E must be the correct answer. Let's check.

Say a = - 3 and b = -2. Thus, from ab < a^2 => -2 *- 3 < (-3)^2 =>6 < 9. This is correct. We can cross check this logically too.

We have ab < a^2 => b > a; cancelling a from both sides, note the sign reversal of the inequality. Since a is negative, upon canceling a, the sign of the inequality will change.

So, we are left with b > a, which is correct as per the information a < b < 0.

The correct answer: E

Hope this helps!

-Jay
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BTGmoderatorDC wrote:If a < b < 0, which of the following must be true?

A. a^2 < b^2
B. b − 10 < a
C. b + a > a
D. ab < b^2
E. ab < a^2

OA E

Source: Veritas Prep

Since a is less than b and both a and b are negative, ab will always be less than a^2 (for example, if a = -2 and b = -1, ab = 2 and a^2 = 4).

Answer: E