Problem 10 - Beginning and Intermediate Algebra

Question

Suppose that \(x\) represents a positive number and \(y\) represents a negative \mathrm{number. Determine whether the given expression represents a positive or } a \text { negative number. } $$ x+|y| $$

Step-by-Step Solution

Answer

The expression is positive.

Step 1: Identify the Signs of Each Term

First, understand the signs of each part of the expression. Here, we know that x represents a positive number and y represents a negative number.

Step 2: Evaluate the Absolute Value of y

Given that y is a negative number, take its absolute value. The absolute value of any number is always positive. Therefore, \(|y|\) will be positive.

Step 3: Determine the Sign of the Sum

Next, add the positive number x to the positive \(|y|\). The sum of two positive numbers is always positive.

Key Concepts

Positive and Negative Numbers

Positive and negative numbers are foundational concepts in algebra. Understanding them is key to solving many mathematical problems.

Positives numbers are greater than zero and are typically written without a sign, like 5, 12.7, or 100.
Negative numbers, on the other hand, are less than zero and always have a minus sign in front, such as -3, -15.5, or -200.

In our example, it is given that x is a positive number and y is a negative number.
To understand whether the final expression will be positive or negative, we need to understand how these positive and negative numbers interact with each other through operations like addition or subtraction.

Absolute Value

Absolute value refers to the distance of a number from zero, regardless of direction. It makes all numbers positive.

This concept is represented by vertical bars: \( |y| \).
For example, the absolute value of -5 is written as \( |-5| \) and equals 5.
Similarly, the absolute value of 8 is \( |8| \) and remains 8. Absolute value function transforms negative values into positive, preserving the magnitude.

In our given expression where y is negative, its absolute value \( |y| \) will be positive irrespective of the value of y.This means if \( y = -3 \), then \( |y| = 3 \). Constructing the expression with positive and absolute values allows easy determination of the sum's sign.

Sum of Numbers

Summing numbers involves combining their values to find a total.
In algebra, we follow specific rules depending on whether the numbers are positive or negative.

The sum of two positive numbers is always positive. The same is true with negative numbers; their sum remains negative.
However, adding a positive and a negative number depends on their absolute values.

For the expression $$$$x + |y|$$$$, where both terms are now positive, the result will be positive.
Here’s why: Adding two positive values consecutively increases the total.
For instance, if \( x = 5 \) and \( y = -3 \), then \( |y| = 3 \). Consequently, \( x + |y| = 5 + 3 = 8 \), which is positive.