Which of the following are solutions to the inequality below? Select all that apply.
$\frac{r}{17}-5 \leq-9$
We are given the linear inequality:
\[\frac{r}{17} - 5 \leq -9\]
We need to determine which values from the set \(\{51, -51, -119, -34\}\) satisfy this inequality. A linear inequality is an inequality that can be written in the form \(ax + b < c\) or similar. Here, we aim to isolate \(r\) to find the range of values that satisfy the condition.
To solve the inequality \(\frac{r}{17} - 5 \leq -9\), we follow these algebraic steps:
\[\frac{r}{17} \leq -9 + 5\]
\[\frac{r}{17} \leq -4\]
\[r \leq -4 \times 17\]
\[r \leq -68\]
Thus, any value of \(r\) that is less than or equal to -68 is a valid solution to the original inequality.
Now, we test each candidate value to see if it satisfies the condition \(r \leq -68\):
Comparing these to the condition, we find that only \(r = -119\) satisfies the inequality.
The value that satisfies the inequality \(\frac{r}{17} - 5 \leq -9\) is \(r = -119\). Thus, the final answer is \(\boxed{r = -119}\).
Imagine you are managing the temperature in a large storage freezer. The temperature \(r\) in a specific cooling unit must be kept at a certain level to preserve sensitive materials. Specifically, the relationship \(\frac{r}{17} - 5 \leq -9\) represents the threshold for safety. By solving this, you identify which temperature readings are within the safe range. In our case, the temperature must be \(-68\) or colder to ensure the items remain frozen, so a reading of \(-119\) degrees is a safe condition.