Problem 14

Explanation

We need to find the value of $x$ that does not satisfy the inequality $\frac{144}{29} < x < 5$.

Analyzing the Lower Bound

First, let's rewrite the lower bound $\frac{144}{29}$ to make it easier to compare with the number 5.

So, the given inequality is:

This means that for $x$ to satisfy the condition, it must lie between $5 - \frac{1}{29}$ and $5$.

Analyzing the Options

We will convert each option into the form $5 - \frac{1}{n}$ and compare them.

1. Analyzing $\frac{139}{28}$

   $$\frac{139}{28} = \frac{140 - 1}{28} = 5 - \frac{1}{28}$$

   Let's compare $5 - \frac{1}{28}$ with the lower bound $5 - \frac{1}{29}$. Since $28 < 29$, we have $\frac{1}{28} > \frac{1}{29}$. Consequently, $5 - \frac{1}{28} < 5 - \frac{1}{29}$.

   This value is smaller than the lower bound, so it does not satisfy the inequality.

2. Analyzing $\frac{149}{30}$

   $$\frac{149}{30} = \frac{150 - 1}{30} = 5 - \frac{1}{30}$$

   Since $30 > 29$, we have $\frac{1}{30} < \frac{1}{29}$. Therefore, $5 - \frac{1}{30} > 5 - \frac{1}{29}$. This value satisfies the inequality.

3. Analyzing $\frac{154}{31}$

   $$\frac{154}{31} = \frac{155 - 1}{31} = 5 - \frac{1}{31}$$

   Since $31 > 29$, this value satisfies the inequality.

4. Analyzing $\frac{159}{32}$

   $$\frac{159}{32} = \frac{160 - 1}{32} = 5 - \frac{1}{32}$$

   Since $32 > 29$, this value satisfies the inequality.

5. Analyzing $\frac{164}{33}$

   $$\frac{164}{33} = \frac{165 - 1}{33} = 5 - \frac{1}{33}$$

   Since $33 > 29$, this value satisfies the inequality.

Conclusion

The only value that is not within the range $\frac{144}{29} < x < 5$ is $\frac{139}{28}$ because it is less than the lower bound.

Thus, the answer is $\frac{139}{28}$.