Question

Remove the brackets from the given expression: \(\frac{3}{4}\left(\frac{1}{2} x+7\right)\)

Short answer

Question: Remove the brackets from the given expression: \(\frac{3}{4}\left(\frac{1}{2}x+7\right)\). Answer: \(\frac{3}{8}x + \frac{21}{4}\).

Step-by-step solution

Identify the terms inside the bracket

The terms inside the bracket are: \(\frac{1}{2}x\) and \(7\).

Multiply first term inside the bracket by the outside fraction

Multiply the first term \(\frac{1}{2}x\) by the outside fractional coefficient \(\frac{3}{4}\). This is done according to the rule \(\frac{a}{b}*\frac{c}{d}=\frac{ac}{bd}\). The resulting fraction is: \(\frac{3}{4}*\frac{1}{2}x = \frac{3*1}{4*2}x = \frac{3}{8}x\).

Multiply second term inside the bracket by the outside fraction

Multiply the second term \(7\) by the outside fractional coefficient \(\frac{3}{4}\). The resulting fraction is: \(\frac{3}{4}*7 = \frac{3*7}{4*1} = \frac{21}{4}\).

Combine the simplified terms

Combine the simplified terms from Steps 2 and 3 to form the final expression without the brackets: \(\frac{3}{4}\left(\frac{1}{2}x+7\right) = \frac{3}{8}x + \frac{21}{4}\).

Key concepts

Fractions

Fractions are expressions that represent parts of a whole.
They can be thought of as a division of two numbers: a numerator and a denominator.
For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.
Understanding fractions is crucial when working with algebraic expressions, especially when they involve operations like multiplication and simplification.

• Types of Fractions: Proper fractions have a numerator smaller than the denominator, such as \(\frac{3}{8}\). Improper fractions have a numerator larger than the denominator, such as \(\frac{9}{4}\).
• Operations: When multiplying fractions, multiply the numerators together and the denominators together. When adding or subtracting, find a common denominator first.
• Simplification: Both the numerator and denominator can sometimes be divided by a greatest common factor to simplify the fraction further.

By mastering fractions, dealing with more complex algebraic expressions becomes easier, especially when using the distributive property.

Distributive Property

The distributive property is a fundamental algebraic principle that involves distributing a single term over the terms inside a bracket.
In simpler terms, it means multiplying each term inside the bracket by the term outside the bracket.
This property helps in removing the brackets from an expression.For instance, in the expression \( \frac{3}{4}(\frac{1}{2}x + 7) \), the distributive property tells us to:

• Multiply \( \frac{3}{4} \) by the first term: \( \frac{1}{2}x \), resulting in \( \frac{3}{8}x \).
• Multiply \( \frac{3}{4} \) by the second term: 7, giving \( \frac{21}{4} \).

All terms are then combined for the final expression.
The distributive property ensures that every term inside the parentheses is accounted for, promoting accurate algebraic manipulations.

Simplifying Expressions

Simplifying expressions makes algebra easier by breaking down complex expressions into simpler, more manageable parts.
This involves performing operations like addition, subtraction, multiplication, and division on the terms.To simplify the expression \( \frac{3}{4}(\frac{1}{2}x + 7) \):

• First, apply the distributive property, as shown in earlier steps, resulting in \( \frac{3}{8}x + \frac{21}{4} \).
• Check if there is any opportunity to combine like terms. In this case, the terms are already simplified since there are no like terms to combine.
• Ensure that all fractions are in their simplest form by looking for common factors.

Simplifying expressions helps in making equations more straightforward to solve and understand.
It relies heavily on understanding of operations and recognizing like terms.