Problem 27 - Fundamentals of Mathematics

Question

For the following 5 problems, what numbers can replace \(m\) so that the following statements are true? \(-7

Step-by-Step Solution

Answer

The integers that can replace \( m \) are -6, -5, -4, -3, and -2.

Step 1: Understand the Problem

We are asked to find values of \( m \) that satisfy the inequality \(-7 < m < -1\), where \( m \) must be an integer.

Step 2: Determine Range of m

The inequality \(-7 < m < -1\) specifies that \( m \) should be greater than -7 and less than -1. Therefore, we need to identify integers between -7 and -1.

Step 3: Identify Integer Values

List integers greater than -7: -6, -5, -4, -3, -2. These are also less than -1 and thus satisfy the inequality.

Step 4: Verify the Solution

Check each integer to ensure they fall within the inequality constraint. Each of the integers -6, -5, -4, -3, and -2 satisfies \(-7 < m < -1\).

Key Concepts

Inequality Problems

When tackling inequality problems, the goal is to find all possible values for a given variable that satisfy the inequality condition. These types of problems often involve identifying a range within which the variable's values fall.
For instance, in the problem we have -7 < m < -1 where we aim to find values of \(m\) such that this inequality holds. This means \(m\) must be greater than -7 and also less than -1. The inequality, with both bounds given, creates a defined solution space or interval.

Solving inequalities requires careful analysis of the given bounds and checking that the solutions, especially when limited to integers, truly satisfy the conditions in place.

• Translate the inequality into a simpler form.
• Identify the range of possible values.
• Ensure solutions fit the criteria of the problem, such as being integers.

Integer Solutions

Integer solutions refer to identifying whole numbers that meet a certain condition or set of conditions. In an inequality problem, integer solutions mean that the variable can only take on whole number values. This is crucial, as integers are numbers without fractions or decimals (e.g., -3, 0, 4).

In our example problem:
1. The bounds are -7 and -1, and we need to identify whole numbers (integers) between them.
2. By systematically counting from -7 upwards until -1, we identify: -6, -5, -4, -3, and -2.

Each found number is an integer solution because:

• They satisfy the inequality \(-7 < m < -1\).
• They are whole numbers within the specified range.

This approach ensures comprehensive checking to not miss any potential solutions.

Range of Integers

Determining the range of integers involves identifying all integer values that a variable can take, considering any given inequality bounds.
For the inequality -7 < m < -1,
the range of integers is specifically the set of integers greater than -7 and less than -1. These boundaries are not included as they are strict (not \( \leq \), but \(<\)).

To find the range of integers:

• Start at the next integer greater than the lower bound (\(-7+1 = -6\)).
• Continue sequentially increasing till reaching the integer just below the upper bound (\(-1-1 = -2\)).
• Thus, the range of integers is: -6, -5, -4, -3, and -2, which fits within the defined bounds and adheres to the integer-only solution requirement.

By organizing numbers sequentially and reviewing boundary conditions, one can precisely determine the correct range of integers for any inequality problem.