Find all x satisfying (7⋅x+6)/(7⋅x−3)<2

We are tasked with solving the inequality:

\[
\frac{7x + 6}{7x - 3} < 2
\]

### Step 1: Eliminate the fraction by multiplying both sides by \(7x - 3\), but we need to be careful about the sign of \(7x - 3\), because if it's negative, we must reverse the inequality.

- Case 1: \(7x - 3 > 0 \implies x > \frac{3}{7}\)
  
  Multiply both sides by \(7x - 3\):

  \[
  7x + 6 < 2(7x - 3)
  \]

  Expand:

  \[
  7x + 6 < 14x - 6
  \]

  Subtract \(7x\) from both sides:

  \[
  6 < 7x - 6
  \]

  Add 6 to both sides:

  \[
  12 < 7x
  \]

  Divide by 7:

  \[
  x > \frac{12}{7}
  \]

  So in this case, \(x > \frac{12}{7}\), but also \(x > \frac{3}{7}\), meaning \(x > \frac{12}{7}\).

- Case 2: \(7x - 3 < 0 \implies x < \frac{3}{7}\)
  
  In this case, multiplying by \(7x - 3\) reverses the inequality:

  \[
  7x + 6 > 2(7x - 3)
  \]

  Expand:

  \[
  7x + 6 > 14x - 6
  \]

  Subtract \(7x\) from both sides:

  \[
  6 > 7x - 6
  \]

  Add 6 to both sides:

  \[
  12 > 7x
  \]

  Divide by 7:

  \[
  x < \frac{12}{7}
  \]

  So in this case, \(x < \frac{3}{7}\), but also \(x < \frac{12}{7}\), meaning \(x < \frac{3}{7}\).

### Step 2: Combine the results

- From Case 1, we found that \(x > \frac{12}{7}\).
- From Case 2, we found that \(x < \frac{3}{7}\).

### Final Solution:

The solution set is:
\[
x < \frac{3}{7} \quad \text{or} \quad x > \frac{12}{7}
\]

Would you like further clarification or details on any step?

Here are 5 related questions for further exploration:
1. What happens if the inequality changes to \(\frac{7x + 6}{7x - 3} \leq 2\)?
2. How do we handle inequalities involving quadratic expressions?
3. What if the inequality were \(\frac{7x + 6}{7x - 3} > 2\)?
4. Can you solve similar inequalities with different coefficients?
5. How do you graph the solution to this inequality on a number line?

**Tip:** Always remember to check the sign of an expression when multiplying both sides of an inequality!

Find all x satisfying (7⋅x+6)/(7⋅x−3) < 2

Explore how to solve the rational inequality (7x + 6)/(7x - 3) < 2. This detailed step-by-step solution discusses key algebraic concepts such as rational inequalities and handling sign changes. Ideal for students in Grades 10-12.

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