How many values of  satisfy both of the following conditions?
(a)  is a multiple of  not necessarily positive.
(b)  is less than .

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

How many values of x satisfy both of the following conditions? (a) x is a multiple of 7 not necessarily positive. (b) x is less than 100.

Learn how to count the number of multiples of 7 less than 100, including negative values. This step-by-step solution involves key concepts from number theory and inequalities, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation