Find accurate and reliable answers to your questions on IDNLearn.com. Discover in-depth answers to your questions from our community of experienced professionals.
Sagot :
To simplify the expression [tex]\(\frac{8}{x+2} - \frac{6}{x+5}\)[/tex], we need to combine the two fractions over a common denominator.
1. Identify the common denominator:
The denominators are [tex]\(x+2\)[/tex] and [tex]\(x+5\)[/tex]. The common denominator will be the product of these two expressions: [tex]\((x+2)(x+5)\)[/tex].
2. Rewrite each fraction with the common denominator:
To rewrite [tex]\(\frac{8}{x+2}\)[/tex] with the common denominator [tex]\((x+2)(x+5)\)[/tex], we multiply the numerator and denominator by [tex]\((x+5)\)[/tex]:
[tex]\[ \frac{8}{x+2} = \frac{8(x+5)}{(x+2)(x+5)} \][/tex]
Similarly, to rewrite [tex]\(\frac{6}{x+5}\)[/tex] with [tex]\((x+2)(x+5)\)[/tex] as the common denominator, we multiply the numerator and denominator by [tex]\((x+2)\)[/tex]:
[tex]\[ \frac{6}{x+5} = \frac{6(x+2)}{(x+2)(x+5)} \][/tex]
3. Combine the rewritten fractions:
Now, both fractions have the same denominator, so we can subtract the numerators directly:
[tex]\[ \frac{8(x+5)}{(x+2)(x+5)} - \frac{6(x+2)}{(x+2)(x+5)} = \frac{8(x+5) - 6(x+2)}{(x+2)(x+5)} \][/tex]
4. Simplify the numerator:
Distribute the constants in the numerator:
[tex]\[ 8(x+5) - 6(x+2) = 8x + 40 - 6x - 12 \][/tex]
Combine like terms:
[tex]\[ 8x + 40 - 6x - 12 = 2x + 28 \][/tex]
5. Express the final simplified form:
The final expression is:
[tex]\[ \frac{2x + 28}{(x+2)(x+5)} = \frac{2(x + 14)}{(x+2)(x+5)} \][/tex]
So, the simplified form of the expression [tex]\(\frac{8}{x+2} - \frac{6}{x+5}\)[/tex] is:
[tex]\[ \boxed{\frac{2(x + 14)}{(x + 2)(x + 5)}} \][/tex]
1. Identify the common denominator:
The denominators are [tex]\(x+2\)[/tex] and [tex]\(x+5\)[/tex]. The common denominator will be the product of these two expressions: [tex]\((x+2)(x+5)\)[/tex].
2. Rewrite each fraction with the common denominator:
To rewrite [tex]\(\frac{8}{x+2}\)[/tex] with the common denominator [tex]\((x+2)(x+5)\)[/tex], we multiply the numerator and denominator by [tex]\((x+5)\)[/tex]:
[tex]\[ \frac{8}{x+2} = \frac{8(x+5)}{(x+2)(x+5)} \][/tex]
Similarly, to rewrite [tex]\(\frac{6}{x+5}\)[/tex] with [tex]\((x+2)(x+5)\)[/tex] as the common denominator, we multiply the numerator and denominator by [tex]\((x+2)\)[/tex]:
[tex]\[ \frac{6}{x+5} = \frac{6(x+2)}{(x+2)(x+5)} \][/tex]
3. Combine the rewritten fractions:
Now, both fractions have the same denominator, so we can subtract the numerators directly:
[tex]\[ \frac{8(x+5)}{(x+2)(x+5)} - \frac{6(x+2)}{(x+2)(x+5)} = \frac{8(x+5) - 6(x+2)}{(x+2)(x+5)} \][/tex]
4. Simplify the numerator:
Distribute the constants in the numerator:
[tex]\[ 8(x+5) - 6(x+2) = 8x + 40 - 6x - 12 \][/tex]
Combine like terms:
[tex]\[ 8x + 40 - 6x - 12 = 2x + 28 \][/tex]
5. Express the final simplified form:
The final expression is:
[tex]\[ \frac{2x + 28}{(x+2)(x+5)} = \frac{2(x + 14)}{(x+2)(x+5)} \][/tex]
So, the simplified form of the expression [tex]\(\frac{8}{x+2} - \frac{6}{x+5}\)[/tex] is:
[tex]\[ \boxed{\frac{2(x + 14)}{(x + 2)(x + 5)}} \][/tex]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.