To solve the problem of finding how many values of $x$ satisfy the given conditions:

Conditions:

• (a) $x$ is a multiple of 7 (not necessarily positive).
• (b) $x$ is less than 100.

Step 1: Represent $x$ as a multiple of 7

Since $x$ is a multiple of 7, we can write $x = 7k$, where $k$ is an integer (positive, negative, or zero).

Step 2: Find the upper limit for $k$

We are given that $x$ must be less than 100, i.e., $x < 100$. Substituting $x = 7k$ into this inequality:

$7k < 100$

Dividing both sides by 7:

$k < \frac{100}{7} \approx 14.2857$

Thus, the largest integer value for $k$ is $k = 14$.

Step 3: Find the lower limit for $k$

Since $x$ can be any multiple of 7 (not necessarily positive), $x$ can also be negative. We have no lower bound on the value of $x$, but since $k$ is an integer, we can consider how low $k$ can go. We simply need to find when $x = 7k$ becomes negative while still less than 100.

So $x = 7k$ must satisfy the inequality $7k < 100$. If $k = -1$, $x = -7$. As $k$ decreases, $x$ continues to decrease indefinitely.

Step 4: Count the valid values of $k$

Since $x = 7k$ can be negative, we need to find the largest and smallest values of $k$ that satisfy the condition $x < 100$. From Step 2, we know the largest integer $k$ is 14.

Since $k$ can be any negative integer, the set of valid values of $k$ extends indefinitely. Therefore, ( \boxed{-7} gives us an final boundaries therefore adj