Question

check true or false for each comparison below. (dok
a. $3 < \sqrt{12} < 4$ \t\ttrue \tfalse
b. $16 < \sqrt{15} < 17$ \ttrue \tfalse
c. $4 < \sqrt{15} < 5$ \t\ttrue \tfalse
d. $3.5 < \sqrt{15} < 4$ \ttrue \tfalse

Explanation:

for A:

Step1: Square all parts

To check \(3 < \sqrt{12} < 4\), square each part: \(3^2 = 9\), \((\sqrt{12})^2 = 12\), \(4^2 = 16\).

Step2: Compare squared values

Since \(9 < 12 < 16\), the original inequality holds. So A is True.

for B:

Step1: Square all parts

For \(16 < \sqrt{15} < 17\), square each part: \(16^2 = 256\), \((\sqrt{15})^2 = 15\), \(17^2 = 289\).

Step2: Compare squared values

But \(15 < 256\), so \(16 < \sqrt{15}\) is false. Thus B is False.

for C:

Step1: Square all parts

For \(4 < \sqrt{15} < 5\), square each part: \(4^2 = 16\), \((\sqrt{15})^2 = 15\), \(5^2 = 25\).

Step2: Compare squared values

Since \(15 < 16\), \(4 < \sqrt{15}\) is false. So C is False.

for D:

Step1: Square all parts

For \(3.5 < \sqrt{15} < 4\), square \(3.5\): \(3.5^2 = 12.25\), \((\sqrt{15})^2 = 15\), \(4^2 = 16\).

Step2: Compare squared values

Since \(12.25 < 15 < 16\), the original inequality holds. So D is True.

Answer:

A. True
B. False
C. False
D. True