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Chapter 7: Problem 34

Add or subtract. Write answer in lowest terms. \(\frac{2 x-6}{4}+\frac{x+5}{6}\)

Short Answer

Expert verified
\frac{2(x-1)}{3}

Step by step solution

01

Identify the Least Common Denominator (LCD)

To add the fractions \(\frac{2x-6}{4} + \frac{x+5}{6}\), identify the least common multiple of the denominators, 4 and 6. The LCD is 12.
02

Adjust Fractions to Have the Same Denominator

Rewrite each fraction with the denominator of 12. To do this, multiply the numerator and denominator of \(\frac{2x-6}{4} \) by 3, and of \(\frac{x+5}{6} \) by 2: \[ \frac{2x-6}{4} = \frac{3(2x-6)}{3 \times 4} = \frac{6x-18}{12} \] \[ \frac{x+5}{6} = \frac{2(x+5)}{2 \times 6} = \frac{2x+10}{12} \]
03

Add the Fractions

Now that the fractions have the same denominator, add the numerators: \[ \frac{6x-18}{12} + \frac{2x+10}{12} = \frac{(6x-18) + (2x+10)}{12} \] This simplifies to \[ \frac{8x-8}{12} \]
04

Simplify the Fraction

Factor out any common factors in the numerator and the denominator: \[ \frac{8x-8}{12} = \frac{8(x-1)}{12} = \frac{2(x-1)}{3} \] Thus, the fraction in its simplest form is \[ \frac{2(x-1)}{3} \]

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Key Concepts

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

least common denominator
When working with fractions, it's crucial to have the same denominator to add or subtract them. This common denominator is called the least common denominator (LCD). The LCD is the smallest number that both of the original denominators divide into without leaving a remainder.
simplifying fractions
Restricting the numerator and the denominator to their simplest form makes a fraction much easier to understand. To simplify a fraction:
  • Identify the greatest common factor (GCF) of both the numerator and the denominator.
  • Divide both the numerator and the denominator by this GCF.
In our example, \[ \frac{8x-8}{12} = \frac{8(x-1)}{12} \] can be further simplified by dividing both parts by 4, resulting in \[ \frac{2(x-1)}{3} \]
algebraic expressions
An algebraic expression is a combination of numbers, variables, and operations. Adding and subtracting these expressions means combining like terms. Ensure every term in each expression has the same variable components. When combined, emembr:
  • Combine the coefficients of similar terms.
  • Keep the variable part intact.
In the exercise above, we combined \[ \frac{6x-18}{12} + \frac{2x+10}{12} = \frac{8x-8}{12} \]

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Most popular questions from this chapter

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{z^{2}-z-6}{z^{2}-2 z-8} \cdot \frac{z^{2}+7 z+12}{z^{2}-9}$$

Connie McNair's boat goes \(12 \mathrm{mph}\). Find the rate of the current of the river if she can go \(6 \mathrm{mi}\) upstream in the same amount of time she can go \(10 \mathrm{mi}\) downstream.

Rewrite each rational expression with the indicated denominator. $$ \frac{a+2 b}{2 a^{2}+a b-b^{2}}=\frac{?}{2 a^{3} b+a^{2} b^{2}-a b^{3}} $$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{p^{2}+4 p-5}{p^{2}+7 p+10} \div \frac{p-1}{p+4}$$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{m^{2}+3 m+2}{m^{2}+5 m+4} \cdot \frac{m^{2}+10 m+24}{m^{2}+5 m+6}$$
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