Let's closely examine each step of John's work and the correct approach to adding these rational expressions. Here is a breakdown of the correct process:
1. Given expressions:
[tex]\[
\frac{2x + 1}{2x - 3} + \frac{3}{10x - 15}
\][/tex]
2. Factor the denominator of the second fraction:
[tex]\[
10x - 15 = 5(2x - 3)
\][/tex]
So, we have:
[tex]\[
\frac{2x + 1}{2x - 3} + \frac{3}{5(2x - 3)}
\][/tex]
3. Create a common denominator, which is [tex]\(5(2x - 3)\)[/tex]:
[tex]\[
\frac{5(2x + 1)}{5(2x - 3)} + \frac{3}{5(2x - 3)}
\][/tex]
4. Combine the fractions by adding their numerators:
[tex]\[
\frac{5(2x + 1) + 3}{5(2x - 3)} = \frac{10x + 5 + 3}{5(2x - 3)}
\][/tex]
Simplifying the numerator:
[tex]\[
\frac{10x + 8}{5(2x - 3)}
\][/tex]
According to John's steps:
- Step 1: Factor the numerators and denominators
[tex]\[
\frac{2 x+1}{2 x-3} + \frac{3}{5(2 x-3)}
\][/tex]
This step is correct.
- Step 2: Create a common denominator of [tex]\(5(2x - 3)\)[/tex]
[tex]\[
\frac{5(2 x+1)}{5(2 x-3)} + \frac{3}{5(2 x-3)}
\][/tex]
This step is correct.
- Step 3: Distribute
[tex]\[
\frac{10 x+5}{10 x-15} + \frac{3}{10 x-15}
\][/tex]
This step is incorrect because the denominator should still be [tex]\(5(2x - 3)\)[/tex], causing an error here.
- Step 4: Add the numerators and denominators (which is incorrect)
[tex]\[
\frac{10 x+8}{20 x-30}
\][/tex]
The mistake occurs here because John incorrectly adds both the numerators and the denominators directly. The correct approach is to keep the common denominator and only add the numerators.
5. Simplify the incorrect result, which might look correct numerically but not logically:
[tex]\[
\frac{2(5 x+4)}{10(2 x-3)} = \frac{5 x+4}{5(2 x-3)}
\][/tex]
Thus, John made an error at Step 3.
