If \(a \geq 30\) and \(b \leq 15\), then which of the following must also be true?

BTGmoderatorLU wrote: ↑

Mon Aug 17, 2020 12:55 pm

Source: Princeton Review

If \(a \geq 30\) and \(b \leq 15\), then which of the following must also be true?

A. \(a - b \leq 45\)
B. \(a - b \leq 15\)
C. \(a - b \geq 15\)
D. \(a + b \leq 45\)
E. \(a + b \geq 45\)

The OA is C

In many of the other choices, we have -b (subtract b)

So let's take: b ≤ 15
And multiply both sides by -1 to get: -b ≥ -15 [aside: Since I multiplied both sides of the inequality by a NEGATIVE number, I had to REVERSE the direction of the inequality symbol]

So we now have the following:
a ≥ 30
-b ≥ -15

If we ADD the two inequalities we get: a + (-b) ≥ 30 + (-15)
Simplify to get: a - b ≥ 15

Answer: C

BTGmoderatorLU wrote: ↑

Mon Aug 17, 2020 12:55 pm

Source: Princeton Review

If \(a \geq 30\) and \(b \leq 15\), then which of the following must also be true?

A. \(a - b \leq 45\)
B. \(a - b \leq 15\)
C. \(a - b \geq 15\)
D. \(a + b \leq 45\)
E. \(a + b \geq 45\)

The OA is C

Solution:

If a = 31 and b = 1, then a - b = 30 and a + b = 32. We see that we can eliminate choices B and E from the correct answer. Now, if a = 51 and b = 1, then a - b = 50 and a + b = 52. We see that we can also eliminate choices A and D from being the correct answer. Therefore, the correct answer, after the process of elimination, is C.

Answer: C