To simplify the expression [tex]\(\frac{4k - 3}{4} + \frac{k + 6}{2}\)[/tex], we follow these steps:
1. Common Denominator: Notice that the fractions have different denominators (4 and 2). To combine them, we need a common denominator. The least common multiple of 4 and 2 is 4.
2. Rewrite the Second Fraction: Rewrite [tex]\(\frac{k + 6}{2}\)[/tex] with the denominator 4:
[tex]\[
\frac{k + 6}{2} = \frac{(k+6) \cdot 2}{2 \cdot 2} = \frac{2(k + 6)}{4}
\][/tex]
3. Combine the Fractions: Rewrite the original expression with the common denominator:
[tex]\[
\frac{4k - 3}{4} + \frac{2(k + 6)}{4}
\][/tex]
Since both fractions now have a common denominator, they can be combined:
[tex]\[
\frac{4k - 3 + 2(k + 6)}{4}
\][/tex]
4. Distribute and Combine Like Terms:
Distribute the 2 in the second term:
[tex]\[
2(k + 6) = 2k + 12
\][/tex]
Substituting back in, we get:
[tex]\[
\frac{4k - 3 + 2k + 12}{4}
\][/tex]
5. Combine Like Terms in the Numerator:
Combine [tex]\(4k\)[/tex] and [tex]\(2k\)[/tex] to get [tex]\(6k\)[/tex], and combine [tex]\(-3\)[/tex] and [tex]\(12\)[/tex] to get [tex]\(9\)[/tex]:
[tex]\[
\frac{6k + 9}{4}
\][/tex]
6. Simplified Expression: Now we separate the fraction:
[tex]\[
\frac{6k + 9}{4} = \frac{6k}{4} + \frac{9}{4}
\][/tex]
7. Reduce the Fraction [tex]\(\frac{6k}{4}\)[/tex]: Simplify [tex]\(\frac{6k}{4}\)[/tex]:
[tex]\[
\frac{6k}{4} = \frac{3k}{2}
\][/tex]
Therefore, the expression can be simplified further as:
[tex]\[
\frac{3k}{2} + \frac{9}{4}
\][/tex]
The final simplified form of the expression [tex]\(\frac{4k - 3}{4} + \frac{k + 6}{2}\)[/tex] is:
[tex]\[
\frac{3k}{2} + \frac{9}{4}
\][/tex]
