Problem 30 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. decrease a number by \(-1\)

Step-by-Step Solution

Answer

The algebraic expression is \( x + 1 \).

Step 1: Identify the Unknown

First, we identify the unknown quantity we need to represent with a variable. Let’s use the variable \( x \) to represent 'a number' in the problem.

Step 2: Create the Expression for Decrease

The problem requires decreasing the number by \(-1\), which means subtracting \(-1\) from our variable. We set up the expression: \( x - (-1) \).

Step 3: Simplify the Expression

Simplifying the expression: \( x - (-1) = x + 1 \). Since subtracting a negative is equivalent to adding the positive, the expression becomes \( x + 1 \).

Key Concepts

Variable Representation

Whenever we encounter an algebra problem, one of the first steps is identifying the variable, which is a symbol that represents an unknown quantity. In our exercise, the phrase "a number" refers to an unknown value. To make this computation easy, we use a variable, most commonly a letter like \( x \), to stand in for this unknown number.
This way, rather than guessing or trying random numbers, we have a constant symbol that represents exactly what we're working with.

• Variables let us write equations or expressions that hold true for any value.
• They provide a simplified way to navigate through a problem, especially when trying to solve or translate phrases into meaningful expressions.

When choosing a variable, there's no strict rule about which letter to use. However, common practice often sees \( x \), \( y \), or \( z \) as the popular choices. Remember, the variable you choose does not change the math you're doing; it's just a placeholder.

Simplifying Expressions

Once an algebraic expression is set up, the next critical step is simplifying it. Simplifying means making the expression as short and uncomplicated as possible. In our exercise, we started with the expression \( x - (-1) \).
But how did we end up with \( x + 1 \)?
Simplifying often involves performing operations like combining like terms or reducing fractions, but here, we dealt with subtracting a negative number.

Simplifying steps can include:

• Recognizing that subtracting a negative number is the same as adding its positive counterpart.
• Understanding each step transforms the expression into a more easily readable form.

Simplifying doesn’t change the value or the meaning of the expression. It just makes solving it clearer and requires fewer steps later. Practice this skill, as it makes tougher problems much more manageable!

Negative Numbers in Algebra

Working with negative numbers can sometimes be tricky, but they follow consistent rules that help simplify expressions and solve equations.
In our exercise, we had to decrease a number by , which translates to subtracting a negative number.Here’s what happens when dealing with negative numbers:

• Subtracting a negative number (like \(-1\)) is the same as adding the positive version of that number.
• This is why \( x - (-1) \) becomes \( x + 1 \).

Understanding how negatives work not only helps in simplifying expressions but is essential in solving equations or inequalities.
They are part of the integral rules of arithmetic and algebra:

• Two negatives make a positive.
• A positive and a negative yield a negative when multiplied or divided.

Mastering these ideas ensures you'll tackle more complex algebraic tasks with ease.