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Chapter 7: Problem 29

Perform the indicated operation. Simplify, if possible. $$ \frac{2}{x+5}+\frac{3}{4 x} $$

Short Answer

Expert verified
\(\frac{t-4}{t+3}\)

Step by step solution

01

Identify the common denominator

Both fractions have the common denominator \(\(t^2+6t+9\)\). Recognize that they can be combined into a single fraction.
02

Combine the fractions

Add the numerators of the fractions while keeping the denominator the same: \(\(\frac{t^2-3t+2t-12}{t^2+6t+9}\).\)
03

Simplify the numerator

Combine like terms in the numerator: \(\(t^2 -3t + 2t - 12 = t^2 - t - 12\).\)So the fraction now is \(\(\frac{t^2 - t - 12}{t^2 + 6t + 9}\).\)
04

Factor both the numerator and the denominator

The numerator can be factored as \(\(t^2 - t - 12 = (t - 4)(t + 3)\).\)The denominator can be factored as \(\(t^2 + 6t + 9 = (t + 3)(t + 3) = (t + 3)^2\).\)
05

Simplify the fraction

Once the numerator and denominator are factored, write them as\(\(\frac{(t-4)(t+3)}{(t+3)(t+3)}\)\).Cancel out the common factor \(\(t+3\)\) in the numerator and denominator. This simplifies to \(\(\frac{t-4}{t+3}\).\)

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Key Concepts

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

Common Denominator
When working with fractions, it's essential to have a common denominator. This allows you to combine them easily. In this exercise, both fractions already share the same denominator, \(t^2 + 6t + 9\). This simplifies our work because we don't need to find a new common denominator.

When fractions have the same denominator,
  • You can add or subtract the numerators directly.
  • The denominator remains unchanged.
Identifying common denominators quickly can save you time and help avoid mistakes.
Simplify Fractions
Simplifying fractions makes them easier to understand and work with. After combining fractions, the next step is simplification. Let's take the expression \(\frac{t^2 - 3t + 2t - 12}{t^2 + 6t + 9}\). You need to:

  • Combine like terms in the numerator.
  • Factor both the numerator and the denominator.
For example, combining like terms in \(t^2 - 3t + 2t - 12\), we get \(t^2 - t - 12\). The simplified fraction looks much cleaner and is easier to understand.
Combine Like Terms
Combining like terms helps simplify expressions. In our example, the terms in the numerator are \(t^2 - 3t + 2t - 12\). You need to group terms with the same variables together:
  • Combine \(t^2\) terms.
  • Combine \(t\) terms: \(-3t + 2t\).
  • Combine constant terms: \(-12\).
After combining like terms, you get a simplified numerator: \(t^2 - t - 12\). Combining like terms is a crucial skill in algebra.
Factoring Polynomials
Factoring polynomials is an important technique in algebra. It involves breaking down a polynomial into simpler components (factors). In our example, you need to factor both the numerator and the denominator:

For the numerator \(t^2 - t - 12\):
  • Find two numbers that multiply to -12 and add to -1 (coefficients of \(t^2\) and \(t\).
  • These numbers are -4 and 3.
  • Write it as \((t - 4)(t + 3)\).
For the denominator \(t^2 + 6t + 9\):
  • Find two numbers that multiply to 9 and add to 6.
  • These numbers are 3 and 3.
  • Write it as \((t + 3)(t + 3) = (t + 3)^2\).
After factoring, the expression \(\frac{(t - 4)(t + 3)}{(t + 3)^2}\) can be simplified further by canceling out common factors. This gives us \(\frac{t - 4}{t + 3}\).

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