Problem 46

Question

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

Step-by-Step Solution

Verified

Answer

The simplified expression is $ 3t - 8 $.

1Step 1: Simplify the Fraction

The expression $ \left( \frac{9}{6} t - 4 \right) 2 $ begins with a fraction inside the brackets. Simplify $ \frac{9}{6} $ by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This results in $ \frac{3}{2} $. Thus, the expression becomes $ \left( \frac{3}{2} t - 4 \right) 2 $.

2Step 2: Distribute the Multiplication

The expression now is $ \left( \frac{3}{2} t - 4 \right) 2 $. Distribute the 2 across the expression inside the brackets: multiply both $ \frac{3}{2} t $ and $-4$ by 2. This gives: $ 2 \times \frac{3}{2} t = 3t $ and $ 2 \times (-4) = -8 $.

3Step 3: Write the Final Expression

Combine the results from the distribution to get the simplified expression. Thus, the expression $ \left( \frac{9}{6} t - 4 \right) 2 $ simplifies to $ 3t - 8 $.

Key Concepts

FractionsDistribution PropertyGreatest Common Divisor

Fractions

When working with algebraic expressions, fractions often appear and need to be simplified. A fraction consists of a numerator (the top number) and a denominator (the bottom number). Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator share no common factors other than 1.

For instance, consider the fraction $ \frac{9}{6} $. To simplify it, we look for the largest number that both 9 and 6 can be divided by, known as the greatest common divisor (GCD). For 9 and 6, the GCD is 3.

Divide the numerator (9) by 3, which gives us 3.
Divide the denominator (6) by 3, which gives us 2.

Thus, $ \frac{9}{6} $ simplifies to $ \frac{3}{2} $. Reducing fractions makes calculations easier and the results more understandable. Always check and see if a fraction can be simplified before moving ahead with more complex operations.

Distribution Property

The distribution property, also known as the distributive law, is a fundamental algebraic property that helps expand expressions. It allows you to multiply a single term by each term within a bracket separately.

Consider the expression $ (\frac{3}{2} t - 4) 2 $. Here, the number 2 outside the brackets needs to be distributed across each term inside the brackets.

Multiply $ \frac{3}{2} t $ by 2, resulting in $ 3t $.
Multiply $-4$ by 2, resulting in $-8$.

Using the distribution property effectively means you distribute correctly and combine like terms if necessary. It often simplifies expressions quickly and helps in solving equations efficiently.

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept when simplifying fractions. It is the largest positive integer that divides two or more numbers without leaving a remainder. Finding the GCD helps reduce fractions to their simplest forms, making calculations easier and expressions clearer.

For instance, when simplifying $ \frac{9}{6} $, determining the GCD is the first step.

List the factors of 9: 1, 3, 9.
List the factors of 6: 1, 2, 3, 6.
The common factors are 1 and 3, with 3 being the largest.

Thus, the GCD of 9 and 6 is 3. By dividing both the numerator and the denominator by their GCD, we simplify the fraction to $ \frac{3}{2} $. Understanding and finding the GCD is a valuable skill, especially in algebra, to simplify and solve various problems effectively.

Problem 46

Problem 47

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