To simplify the given expression
[tex]\[
\frac{7-\frac{5}{3 w}}{\frac{5}{3 w}-2}
\][/tex]
we can follow these steps:
### Step 1: Define the problem
We start with the given expression:
[tex]\[
\frac{7-\frac{5}{3 w}}{\frac{5}{3 w}-2}
\][/tex]
### Step 2: Simplify the Numerator and Denominator
First, let's work on simplifying the numerator and the denominator separately.
- Numerator (N):
[tex]\[
7 - \frac{5}{3w}
\][/tex]
- Denominator (D):
[tex]\[
\frac{5}{3w} - 2
\][/tex]
### Step 3: Common Denominator in the Numerator
To simplify the numerator, we want to combine the terms into a single fraction. Notice that the terms [tex]\(7\)[/tex] and [tex]\(\frac{5}{3w}\)[/tex] have different forms.
To combine them, we can express 7 with the common denominator [tex]\(3w\)[/tex]:
[tex]\[
7 = \frac{21w}{3w}
\][/tex]
Thus, the numerator becomes:
[tex]\[
7 - \frac{5}{3w} = \frac{21w}{3w} - \frac{5}{3w} = \frac{21w - 5}{3w}
\][/tex]
### Step 4: Common Denominator in the Denominator
Similarly, for the denominator:
[tex]\[
\frac{5}{3w} - 2
\][/tex]
We can express 2 with the common denominator [tex]\(3w\)[/tex]:
[tex]\[
2 = \frac{6w}{3w}
\][/tex]
Thus, the denominator becomes:
[tex]\[
\frac{5}{3w} - 2 = \frac{5}{3w} - \frac{6w}{3w} = \frac{5 - 6w}{3w}
\][/tex]
### Step 5: Combine the Simplified Numerator and Denominator
Our fraction now looks like this:
[tex]\[
\frac{\frac{21w - 5}{3w}}{\frac{5 - 6w}{3w}}
\][/tex]
### Step 6: Simplify the Complex Fraction
We can simplify this complex fraction by multiplying by the reciprocal of the denominator:
[tex]\[
\frac{\frac{21w - 5}{3w}}{\frac{5 - 6w}{3w}} = \frac{21w - 5}{3w} \times \frac{3w}{5 - 6w}
\][/tex]
The [tex]\(3w\)[/tex] terms cancel out, simplifying the expression to:
[tex]\[
\frac{21w - 5}{5 - 6w}
\][/tex]
### Step 7: Conclusion
Thus, the simplified form of the given expression
[tex]\[
\frac{7-\frac{5}{3 w}}{\frac{5}{3 w}-2}
\][/tex]
is:
[tex]\[
\boxed{\frac{5 - 21w}{6w - 5}}
\][/tex]
Note that we have factored and rearranged terms to achieve a more compact form.
