Problem 25 - Algebra for College Students

Question

Determine whether each is an expression or an equation. Simplify any expressions, and solve any equations. $$ \frac{1}{2} x-\frac{1}{6} x+\frac{3}{2}-8 $$

Step-by-Step Solution

Answer

\( \frac{1}{3} x - \frac{13}{2} \)

Step 1 - Identify the problem

First, recognize whether the given mathematical statement is an expression or an equation. An expression does not have an equal sign, while an equation has an equal sign.

Step 2 - Verify absence of equal sign

Review the given statement: \( \frac{1}{2} x - \frac{1}{6} x + \frac{3}{2} - 8 \). There is no equal sign, so this is an expression.

Step 3 - Simplify the expression

Combine like terms to simplify the expression:Step 3.1 - Combine the terms involving \( x \): \( \frac{1}{2} x - \frac{1}{6} x \) Convert fractions to a common denominator (6): \[ \frac{3}{6} x - \frac{1}{6} x = \frac{2}{6} x = \frac{1}{3} x \]Step 3.2 - Combine the constants: \( \frac{3}{2} - 8 \) Convert 8 to a fraction with a denominator of 2: \[ \frac{3}{2} - \frac{16}{2} = \frac{3-16}{2} = \frac{-13}{2} \]

Step 4 - Write the simplified expression

Combine the simplified \( x \) term and the constant term: \( \frac{1}{3} x - \frac{13}{2} \)

Key Concepts

algebraic expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division) but without an equal sign. In other words, it's a mathematical phrase that can contain constants, coefficients, and variables.
An example is: \(\frac{1}{2} x - \frac{1}{6} x + \frac{3}{2} - 8\). This is an algebraic expression because it lacks an equal sign.

Key features:

• **Variables**: Letters that represent unknown values (like x in our example).
• **Constants**: Fixed numbers (e.g., -8).
• **Coefficients**: Numbers that multiply the variables (e.g., \(\frac{1}{2}\) and \(\frac{1}{6}\) for x).
• **Operations**: Such as addition, subtraction, multiplication, and division.
• **Terms**: Parts of the expression separated by addition or subtraction signs.

Understanding these components will help you identify and work with algebraic expressions more effectively.

combining like terms

One of the easiest ways to simplify algebraic expressions is by combining like terms. Like terms have the same variable raised to the same power.
In the expression \(\frac{1}{2} x - \frac{1}{6} x + \frac{3}{2} - 8\), we have like terms that involve 'x' and constants:

• **Terms involving x**: \(\frac{1}{2} x - \frac{1}{6} x\).
• **Constants**: \(\frac{3}{2}\) and -8.

To combine these:

• Make sure the fractions involved have a common denominator.
• Combine the coefficients (numbers in front of the variables).
• Combine constants.

For example,
\(\frac{1}{2} x\) becomes \(\frac{3}{6} x\),
so \(\frac{1}{2} x - \frac{1}{6} x = \frac{3}{6} x - \frac{1}{6} x = \frac{2}{6} x = \frac{1}{3} x\).
The constants \( \frac{3}{2} \) and -8 need to be combined next:\

fractions

Fractions are a fundamental part of algebra and can represent parts of a whole. They are written as \( \frac{a}{b} \), where 'a' is the numerator, and 'b' is the denominator.
Key points to remember when working with fractions:

• **Common denominators**: To add or subtract fractions, they must have the same denominator.
• **Simplifying fractions**: Reduce fractions to their simplest form by dividing the numerator and the denominator by their greatest common divisor.
• **Converting**: Convert fractions to a common denominator before combining.

In our example:
\( \frac{1}{2} \) changes to \(\frac{3}{6}\) to match \( \frac{1}{6} \). This step makes combining terms easier and more intuitive. For the constants, -8 converts to \( \frac{-16}{2} \) so you can subtract it from \( \frac{3}{2} \) directly.

Remember, fractions are a handy tool in algebra. Understanding how to convert and simplify them makes dealing with complex expressions much easier.