Let's start by analyzing the given expression:
[tex]\[
\frac{4x + 3}{2x} + \frac{3}{5}
\][/tex]
First, let's combine the two separate terms under a common denominator. To do this, we need a common denominator for [tex]\(2x\)[/tex] and [tex]\(5\)[/tex], which is [tex]\(10x\)[/tex].
To combine the fractions, we'll convert each term to have a denominator of [tex]\(10x\)[/tex]:
[tex]\[
\frac{4x + 3}{2x} = \frac{(4x + 3) \cdot 5}{2x \cdot 5} = \frac{20x + 15}{10x}
\][/tex]
Next, convert [tex]\(\frac{3}{5}\)[/tex] to have a denominator of [tex]\(10x\)[/tex]:
[tex]\[
\frac{3}{5} = \frac{3 \cdot 2x}{5 \cdot 2x} = \frac{6x}{10x}
\][/tex]
Now, we add these two fractions:
[tex]\[
\frac{20x + 15}{10x} + \frac{6x}{10x} = \frac{20x + 15 + 6x}{10x} = \frac{26x + 15}{10x}
\][/tex]
We now have the expression in the form:
[tex]\[
\frac{26x + 15}{10x}
\][/tex]
Thus, comparing this with the form [tex]\(\frac{cx + b}{cx}\)[/tex], we can see that [tex]\(c = 10\)[/tex], [tex]\(b = 15\)[/tex], and the simplified expression is indeed:
[tex]\[
\frac{26x + 15}{10x}
\][/tex]
Therefore, the constants [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are as follows:
- [tex]\(a = 10\)[/tex] (denominator coefficient for [tex]\(x\)[/tex])
- [tex]\(b = 15\)[/tex] (constant term in the numerator)
- [tex]\(c = 26\)[/tex] (numerator coefficient for [tex]\(x\)[/tex])
Note that [tex]\(a\)[/tex] is the coefficient of [tex]\(x\)[/tex] in the denominator, and [tex]\(b\)[/tex] remains as the constant term in the numerator. Finally, the simplified form of the given expression is:
[tex]\[
\frac{26x + 15}{10x}
\][/tex]
