Math Problem Statement

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

Solution

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

Conditions:

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

Step 1: Represent xx as a multiple of 7

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

Step 2: Find the upper limit for kk

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

7k<1007k < 100

Dividing both sides by 7:

k<100714.2857k < \frac{100}{7} \approx 14.2857

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

Step 3: Find the lower limit for kk

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

So x=7kx = 7k must satisfy the inequality 7k<1007k < 100. If k=1k = -1, x=7x = -7. As kk decreases, xx continues to decrease indefinitely.

Step 4: Count the valid values of kk

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

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

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Math Problem Analysis

Mathematical Concepts

Number Theory
Inequalities

Formulas

x = 7k, where k is an integer
7k < 100

Theorems

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Suitable Grade Level

Grades 8-10