Q10. - Algebra 2

Question

If n is a prime integer such that $2n>19\ge \frac{7}{8}n$ what is one possible value of n?

Step-by-Step Solution

Answer

The one possible value for n is 11. 

Step-1 – Apply the concept of prime integers

The set of integers are the positive and negative whole numbers including 0 and is expressed as $\dots ,-2,-1,0,1,2,\dots$.

The set of prime numbers are those numbers which have only two factors that are 1 and number itself.

Step-2 –Solve the inequality

Consider the provided inequality $2n>19\ge \frac{7}{8}n$. 

Divide the inequality by n.

$\begin{array}{c}\frac{2n}{n}>\frac{19}{n}\ge \frac{7}{8}\frac{n}{n}\\ 2>\frac{19}{n}\ge \frac{7}{8}\end{array}$

Take the reciprocal and reverse the order of inequality.

$\begin{array}{c}\frac{2n}{n}>\frac{19}{n}\ge \frac{7}{8}\frac{n}{n}\\ 2>\frac{19}{n}\ge \frac{7}{8}\\ \frac{1}{2}<\frac{n}{19}\le \frac{8}{7}\end{array}$

Step-3 &ndash;Simplify the inequality

Multiply the inequality by 19.

$\begin{array}{c}\frac{1}{2}\cdot 19<\frac{n}{19}\cdot 19\le \frac{8}{7}\cdot 19\\ \frac{19}{2}<n\le \frac{152}{7}\end{array}$

Convert to decimal notation.

$9.5<n\le 21.71$

Now, the prime integers that lie between 9.5 and 21.71 are possible values of n.

Therefore, one prime integer is 11. Because only factors of 11 are 1 and 11 only.

Thus, the one possible value for n is 11.