Problem 60 - Algebra 1

Question

Complete the statement using \(>,<, \geq,\) or \(\leq\). If \(m \leq 8\) then_8$=m.

Step-by-Step Solution

Answer

The correct symbol to complete the statement is \( \leq \). The completed statement is: If \( m \leq 8 \) then \( m \leq 8 \).

Step 1: Identify the Given Inequality Operator

In the given statement, the inequality operator is \( \leq \), which stands for 'less than or equal to'. So, \( m \leq 8 \) means \( m \) is either less than 8 or equal to 8.

Step 2: Evaluate the Required Inequality

Given that \( 8 = m \), by reversing this equation, we can conclude that \( m = 8 \). This is a case where \( m \) is equal to 8, which is a possibility considering that \( m \leq 8 \). Therefore, based on the provided condition, \( m \leq 8 \) and \( m = 8 \), the suitable operator to fill the blank is \( \leq \).

Step 3: Fill in the blank with the correct operator

Replacing the blank with the correct operator, the completed statement becomes: If \( m \leq 8 \) then \( m \leq 8 \).

Key Concepts

Inequality Symbols

Inequality symbols are crucial in mathematics for expressing the relationship between two values, indicating if one is larger, smaller, or has a potentially equal relationship to the other. The primary inequality symbols include:

• \( > \): Greater than
• \( < \): Less than
• \( \geq \): Greater than or equal to
• \( \leq \): Less than or equal to

These symbols help us compare quantities effectively. For example, when we say \( m \leq 8 \), it tells us that the number \( m \) is either less than or equal to 8. This means \( m \) can take any value from the possible range up to and including 8. This inclusive nature is important in many real-world scenarios where flexibility is needed in constraints or ranges.

Less Than or Equal To

The symbol \( \leq \) represents "less than or equal to". It is used to show a relationship where one value can either be smaller than or exactly equal to another value.
When we say \( m \leq 8 \), this means:

• \( m \) is any number less than 8, such as 7, 6, 5, etc.
• Or, \( m \) can be exactly 8.

This dual meaning makes \( \leq \) a versatile operator. It allows us to include the endpoint value in potential solutions. This is common in various mathematical and practical applications, such as defining the range of a function or setting boundaries within which variables must stay.
Understanding this symbol is essential for correctly interpreting and solving inequalities.

Inequality Solutions

Solving inequalities is about finding the set of values that satisfy the inequality condition. Given an inequality like \( m \leq 8 \), finding the solution involves:

• Identifying all values of \( m \) that are less than or equal to 8.
• Including the smallest possible number to the greatest permissible number, i.e., from the minimum integer smaller than or equal to 8, continuing indefinitely downwards until a boundary conditions or stopping point is defined.

In our specific example, if \( m \) must satisfy \( m \leq 8 \), acceptable solutions include any number like 8, 7, 6, 5, and so on without a negative constraint. Solutions help us define a feasible set or region where our variable can lie.
Ensuring we read inequalities correctly can drastically change our result interpretation, especially in algebraic problem-solving. Therefore, our solution should always reflect the detailed examination of inequality conditions like whether equal or strictly bounded solutions are needed.