Sure, let's solve the inequality step-by-step.
### Step 1: Write down the inequality
[tex]\[
\frac{1}{3z - 7} < 2
\][/tex]
### Step 2: Eliminate the fraction
To eliminate the fraction, we can multiply both sides by the square of the denominator [tex]\((3z - 7)^2\)[/tex]. This step requires us to consider the sign of the denominator, since multiplying by negative values would reverse the inequality.
#### Consideration A: [tex]\(3z - 7 > 0\)[/tex]
If [tex]\(3z - 7 > 0\)[/tex], then [tex]\(3z > 7\)[/tex] or [tex]\(z > \frac{7}{3}\)[/tex]. In this scenario:
1. Multiply both sides of the inequality by [tex]\((3z - 7)\)[/tex]:
[tex]\[
1 < 2(3z - 7)
\][/tex]
2. Distribute the 2 on the right side:
[tex]\[
1 < 6z - 14
\][/tex]
3. Add 14 to both sides to isolate the term involving [tex]\(z\)[/tex]:
[tex]\[
15 < 6z
\][/tex]
4. Divide both sides by 6:
[tex]\[
\frac{15}{6} < z
\][/tex]
5. Simplify the fraction:
[tex]\[
2.5 < z
\][/tex]
So, [tex]\(z > 2.5\)[/tex].
#### Consideration B: [tex]\(3z - 7 < 0\)[/tex]
If [tex]\(3z - 7 < 0\)[/tex], then [tex]\(3z < 7\)[/tex] or [tex]\(z < \frac{7}{3}\)[/tex]. In this scenario, the inequality will reverse:
1. Multiply both sides of the inequality by [tex]\((3z - 7)\)[/tex]:
[tex]\[
1 < 2(3z - 7)
\][/tex]
2. Distribute the 2 on the right side:
[tex]\[
1 < 6z - 14
\][/tex]
3. Add 14 to both sides to isolate the term involving [tex]\(z\)[/tex]:
[tex]\[
15 < 6z
\][/tex]
4. Divide both sides by 6:
[tex]\[
\frac{15}{6} < z
\][/tex]
5. Simplify the fraction:
[tex]\[
2.5 < z
\][/tex]
So, [tex]\(z > 2.5\)[/tex].But remember, the initial condition was [tex]\(z < \frac{7}{3}\)[/tex]. Therefore there is no suitable value for z which will fit in all the in-qualities mentioned above.
### Conclusion
Combining both cases, the inequality simplifies to:
[tex]\[
z > 2.5
\][/tex]
Thus, the solution to the inequality [tex]\(\frac{1}{3z - 7} < 2\)[/tex] is:
[tex]\[
z > 2.5
\][/tex]
