Question

The number of integers satisfying the inequality \(|n^{2}-100|<50\) is

• A. 5
• B. 6
• C. 12
• D. 8
• E. 10

Hint:

Number Theory Tip: When asked for "integers" satisfying an equation with an even exponent (like $n^2$ or $n^4$), never forget to count the negative roots!

Answer 1

Answer: (E)

Concept:
An absolute value inequality of the form $|X| < a$ (where $a > 0$) translates to the compound inequality $-a < X < a$. By expanding this, we can find the continuous range of values for $X$ and subsequently count the discrete integers that fall within it.

Step 1: Expand the absolute value inequality.
Given the inequality $|n^2 - 100| < 50$, apply the absolute value property to remove the bars: $$-50 < n^2 - 100 < 50$$

Step 2: Isolate the squared variable.
Add 100 to all three parts of the compound inequality to isolate $n^2$: $$-50 + 100 < n^2 < 50 + 100$$ $$50 < n^2 < 150$$

Step 3: Identify the perfect squares within the range.
We need to find integers whose squares are strictly between 50 and 150. Let's list the squares of positive integers: $7^2 = 49$ (Too small) $8^2 = 64$ (Valid) $9^2 = 81$ (Valid) $10^2 = 100$ (Valid) $11^2 = 121$ (Valid) $12^2 = 144$ (Valid) $13^2 = 169$ (Too large)

Step 4: List the valid positive and negative integers.
The valid magnitudes for $n$ are 8, 9, 10, 11, and 12. Since squaring a negative integer yields the same positive result (e.g., $(-8)^2 = 64$), we must include the negative counterparts: $$n \in \{\pm 8, \pm 9, \pm 10, \pm 11, \pm 12\}$$

Step 5: Count the total number of integers.
There are exactly 5 positive integers and 5 negative integers in our set. Total valid integers = $5 + 5 = 10$. Hence the correct answer is (E) 10.