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Chapter 7: Problem 56

Perform the indicated operations. $$ \frac{x+3}{4}-\frac{2 x-1}{4} $$

Short Answer

Expert verified
The simplified expression is \(\frac{-x + 4}{4}\).

Step by step solution

01

Identifying Like Denominators

Notice that both fractions have the same denominator, which is 4. This means you can combine them by simply subtracting the numerators.
02

Performing the Subtraction

Subtract the numerators of the fractions: \[ (x+3) - (2x-1) \]Make sure to distribute the negative sign across the second set of parentheses.
03

Simplifying the Numerator

Simplify the expression in the numerator:\[ x + 3 - 2x + 1 \]Combine like terms to further simplify:\[ -x + 4 \]
04

Writing the Final Result

The final simplified expression of the subtraction is:\[ \frac{-x + 4}{4} \]This is your answer in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Denominators
When subtracting fractions, it is crucial to ensure that the denominators, or bottom numbers, of the fractions are the same. These are known as like denominators. Having like denominators simplifies the process of combining fractions, because you can work directly with the numerators, or top numbers, without any additional steps.
In the given exercise, both fractions have the denominator of 4. This allows you to subtract the fractions right away by focusing only on the numerators:
  • Numerator of the first fraction: \( x+3 \)
  • Numerator of the second fraction: \( 2x-1 \)
You proceed by subtracting these numerators while keeping the denominator constant. Ensuring like denominators first is a fundamental step in working with fractions.
Simplifying Expressions
Simplifying expressions involves reducing them to their cleanest and most straightforward form. This process often includes distributing negative signs and combining numbers and variables where possible.
In the problem at hand, after identifying that the denominators are the same, you proceed to subtract the numerators. This involves:
  • Distributing the negative sign across the second set of parentheses, turning \(- (2x-1)\) into \(-2x+1\)
  • Combining this with the first set, so the expression \( x+3 -2x +1 \) is obtained
Simplifying involves adjusting and reducing this expression to its simplest form by combining terms. This practice makes mathematical expressions easier to interpret and use in further calculations.
Combining Like Terms
The process of combining like terms involves putting together terms that have the same variable or constants. It is an invaluable skill for simplifying algebraic expressions.
In this exercise, once you have distributed the negative sign within the numerators, you get an expression: \( x + 3 - 2x + 1 \). Like terms in this case are:
  • Terms with \( x \): \( x \) and \( -2x \)
  • Constant terms: \( 3 \) and \( 1 \)
To combine like terms:
  • Add or subtract coefficients of terms that share the same variable, in this case, \( x - 2x = -x \)
  • Add constant numbers, leading to \( 3+1 = 4 \)
After combining these like terms, the simplified expression becomes \( -x + 4 \). This step is key for yielding a clear and concise final result in algebraic operations.

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