Problem 128 - Beginning and Intermediate Algebra

Question

If \(q\) is a negative number, \(r\) is a negative number, and t is a positive number, determine whether each expression simplifies to a positive or negative number. If it is not possible to determine, state so. $$ t+r $$

Step-by-Step Solution

Answer

The sign of \(t + r\) cannot be determined without knowing the magnitudes of \(t\) and \(r\).

Step 1: Identify the Signs of the Variables

We are given that \(q\) is negative, \(r\) is negative, and \(t\) is positive. Hence, \(r < 0\) and \(t > 0\).

Step 2: Analyze the Expression

The expression we are given is \(t + r\). We need to determine if this results in a positive number, a negative number, or if more information is required.

Step 3: Combine the Signs in the Expression

Adding a positive number \(t\) to a negative number \(r\) results in an expression whose sign depends on the magnitudes of \(t\) and \(r\).

Step 4: Conclusion on the Sign of the Expression

Since \(t\) is positive and \(r\) is negative, whether \(t + r\) is positive or negative depends on whether \(|t| > |r|\) or \(|r| > |t|\). If \(t > |r|\), the result is positive; if \(|r| > t\), the result is negative; otherwise, it exactly zero.

Step 5: Final Result

It's not possible to definitively determine a sign for \(t + r\) with the given information; we need the magnitudes to conclude.

Key Concepts

Positive and Negative Numbers

Understanding positive and negative numbers is crucial in algebra. A positive number is greater than zero and is situated to the right of zero on the number line. Examples of positive numbers include 1, 2, and 3. They denote an increase or accumulation. Negative numbers, on the other hand, are less than zero and are placed to the left of zero on the number line. Examples are -1, -2, and -3, signaling a decrease or debt, for example.
Adding and subtracting these numbers involve different operations depending on their signs.

• Adding two positive numbers results in a positive sum.
• Adding two negative numbers leads to a negative sum.
• Adding a positive number to a negative number requires comparing their magnitudes to determine the sign of the sum.

Hence, when working with various signs, keeping track of these rules helps in computing correct outcomes. Remember, the position and sign of numbers directly influence results in algebraic expressions.

Magnitude Comparison

Magnitude comparison is a method used to determine which of two numbers has a greater absolute value, regardless of their signs. The magnitude (or absolute value) of a number is its distance from zero on the number line. It's written with vertical bars around the number, like this: \(|x|\).

• The absolute value of a positive number is the number itself. For example, \(|5| = 5\).
• For negative numbers, the absolute value is the positive equivalent. For instance, \(|-4| = 4\).

When comparing magnitudes, you ignore the signs and focus only on their absolute values. In the context of expressions like \(t + r\), if \(|t| > |r|\), then \(t\) 'dominates', and the sum becomes positive. Conversely, if \(|r| > |t|\), \(r\) 'dominates', making the sum negative. If they are equal, the result is zero. Remember, comparing magnitudes allows you to predict the sum's sign when dealing with mixed-signed numbers.

Expression Evaluation

Expression evaluation involves systematically determining the result of an algebraic expression. It requires understanding operations and the interaction between different numbers based on their signs. The exercise provided a good example: determining the sign of \(t + r\) where \(t\) is positive, and \(r\) is negative.

• Step 1: Recognize each number's sign in the expression.
• Step 2: Consider the magnitudes' relationship, affecting the final outcome.

If \(t\) has a greater magnitude than \(r\), \(t + r\) results positively. If \(r\) dominates, the expression evaluates negatively. If their magnitudes are equal, they cancel each other out, resulting in zero.
For effective solution finding, practice spotting how different numbers within an expression interact. It includes recognizing when you need more information, such as the exact values, to arrive at the definitive outcome for the expression's evaluation.