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Chapter 9: Problem 22

Reduce each of the following fractions as completely as possible. $$\frac{4 x+8}{12 x-24}$$

Short Answer

Expert verified
\(\frac{x + 2}{3(x - 2)}\)

Step by step solution

01

Factor the Numerator

Factor the numerator by taking out the greatest common factor (GCF). The GCF of 4 and 8 is 4. So, factor out 4 from the numerator: \[4x + 8 = 4(x + 2)\]
02

Factor the Denominator

Factor the denominator by taking out the greatest common factor (GCF). The GCF of 12 and 24 is 12. So, factor out 12 from the denominator: \[12x - 24 = 12(x - 2)\]
03

Rewrite the Fraction

Substitute the factored forms of the numerator and the denominator back into the fraction: \[\frac{4(x + 2)}{12(x - 2)}\]
04

Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their GCF, which is 4. \[\frac{4(x + 2)}{12(x - 2)} = \frac{4}{12} \times \frac{x + 2}{x - 2} = \frac{1}{3} \times \frac{x + 2}{x - 2} = \frac{x + 2}{3(x - 2)}\] Thus, the reduced fraction is: \[\frac{x + 2}{3(x - 2)}\]

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Key Concepts

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

Factoring Expressions
Factoring expressions involves breaking down a complex expression into simpler factors. This makes operations easier and helps in simplifying fractions, solving equations, and more.
To factor an expression:
  • Identify the greatest common factor (GCF).
  • Extract the GCF from all terms in the expression.
  • Rewrite the expression as the product of the GCF and the simplified terms.
For example, in the exercise: \(4x + 8\), the GCF of 4 and 8 is 4.
So, \(4x + 8 = 4(x + 2)\).
Practice finding the GCF and factoring similar expressions to get more comfortable with this process.
Greatest Common Factor (GCF)
The greatest common factor (GCF) is the largest number that divides all terms in an expression evenly.
It is crucial to find the GCF when simplifying fractions or factoring expressions.
Steps to find the GCF:
  • List out all factors of each term in the expression.
  • Find the largest factor that is common to each term. For the numerator \(4x + 8\), the factors of 4 are 1, 2, and 4, and the factors of 8 are 1, 2, 4, and 8. Therefore, the GCF is 4. For the denominator \(12x - 24\), the factors of 12 are 1, 2, 3, 4, 6, and 12, and the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Thus, the GCF is 12.
Always double-check the GCF to ensure accuracy in your calculations.
Algebraic Simplification
Algebraic simplification is the process of reducing expressions to their simplest form.
It makes expressions easier to work with and understand.
Simplifying involves:
  • Factoring out the GCF in both the numerator and the denominator.
  • Dividing both the numerator and the denominator by their GCF.
In the exercise, we factored out the GCF in both the numerator and the denominator:
\(4(x + 2) \text{ and } 12(x - 2)\).
We then simplified the fraction: \[\frac{4(x + 2)}{12(x - 2)} = \frac{1}{3} \cdot \frac{x + 2}{x - 2} = \frac{x + 2}{3(x - 2)}\].
Simplifying fractions helps in solving algebraic equations more efficiently. Practice this skill to enhance your understanding and problem-solving abilities.

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Most popular questions from this chapter

Find the value of \(c\) so that the line passing through the points \((1, c)\) and \((-5,3)\) is parallel to the line passing through the points \((4,3)\) and \((7,-2)\)

Perform the indicated operations and simplify as completely as possible. $$\left(\frac{c}{2}+\frac{c}{5}\right) \div \frac{c^{2}+7 c}{10}$$

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. $$\left.\frac{x-\mu}{s}<2 \text { for } \mu \quad \text { (Assume } s>0\right)$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{5}{x^{2}-2 x-3}=\frac{4}{x^{2}-3 x-4}$$

Perform the indicated operations and simplify as completely as possible. $$\left(\frac{c}{2} \cdot \frac{c}{5}\right) \div \frac{c^{2}+7 c}{10}$$
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