Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the expression. \(\left(\frac{9}{6} t-4\right) 2\)

Step-by-Step Solution

Answer

The simplified expression is \( 3t - 8 \).

Step 1: Simplify the Fraction

The first step is to simplify the fraction \( \frac{9}{6} \). To simplify it, find the greatest common divisor of the numerator and the denominator, which is 3. Divide both the numerator and the denominator by 3: \( \frac{9}{6} = \frac{9 \div 3}{6 \div 3} = \frac{3}{2} \).

Step 2: Distribute the Number

Now that we have simplified the fraction, we can distribute the number 2 to the terms inside the parenthesis. Rewrite the expression as: \( 2 \left( \frac{3}{2}t - 4 \right) \).

Step 3: Multiply the Terms

Distribute the 2 to each term inside the parenthesis: - Multiply 2 and \( \frac{3}{2} t \): \( 2 \cdot \frac{3}{2} t = 3t \) because \( 2 \cdot \frac{3}{2} = 3 \). - Multiply 2 with -4: \( 2 \cdot -4 = -8 \). So, the expression becomes \( 3t - 8 \).

Step 4: Write the Final Simplified Expression

Combine the results from Step 3 to write the final simplified expression. The simplified expression is \( 3t - 8 \).

Key Concepts

Fraction Simplification

Fraction simplification is a straightforward yet powerful tool in algebra and arithmetic. It involves reducing a fraction to its simplest form.
This means that both the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1.
To simplify a fraction like \( \frac{9}{6} \), start by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both without leaving a remainder. In this example, the GCD is 3.

• Divide the numerator by the GCD: \( 9 \div 3 = 3 \)
• Divide the denominator by the GCD: \( 6 \div 3 = 2 \)

After simplification, \( \frac{9}{6} \) becomes \( \frac{3}{2} \). Simplifying fractions is crucial because it makes calculations easier and expressions cleaner.

Distributive Property

The distributive property is a vital concept in algebra that allows you to distribute a multiplier across terms within parentheses.
Simply put, it helps expand expressions and makes calculation more manageable.
In mathematical terms, it says \( a(b + c) = ab + ac \). This property ensures that you multiply each term inside the bracket by the factor outside.
Let's use our example from the original problem: once we have \( 2 \left( \frac{3}{2}t - 4 \right) \), the multiplier 2 is distributed to each term:

• First, 2 multiplies \( \frac{3}{2}t \): \( 2 \times \frac{3}{2}t = 3t \).
• Next, 2 multiplies -4: \( 2 \times -4 = -8 \).

So, the expression transforms from \( 2 \left( \frac{3}{2}t - 4 \right) \) to \( 3t - 8 \). Understanding the distributive property is essential for simplifying and solving algebraic expressions effectively.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations.
They are fundamental building blocks in algebra and appear frequently in various mathematical problems. The goal often is to simplify these expressions, which means making them as concise as possible without changing their value.
In the given exercise, \( \left( \frac{9}{6} t - 4 \right) 2 \), we see an expression containing a fraction and a variable. To simplify such an expression:

• First, simplify any fractions.
• Then, apply algebraic properties, such as the distributive property.
• Finally, combine like terms if necessary.

By following these steps, the expression is transformed and simplified into \( 3t - 8 \). Mastery of handling algebraic expressions is key for success in algebra and beyond, as it lays the foundation for more complex equations and functions.