To solve the compound inequality 

\[
3x - 8 < 23 \quad \text{OR} \quad -4x + 26 > 6,
\]

we will solve each inequality separately and then combine the results.

### Solving the first inequality: \(3x - 8 < 23\)

1. Add 8 to both sides:

   \[
   3x < 23 + 8,
   \]

   \[
   3x < 31.
   \]

2. Divide by 3:

   \[
   x < \frac{31}{3},
   \]

   \[
   x < 10.33.
   \]

### Solving the second inequality: \(-4x + 26 > 6\)

1. Subtract 26 from both sides:

   \[
   -4x > 6 - 26,
   \]

   \[
   -4x > -20.
   \]

2. Divide by -4 (remember to flip the inequality sign when dividing by a negative number):

   \[
   x < \frac{-20}{-4},
   \]

   \[
   x < 5.
   \]

### Combined Solution:

Since the inequalities are connected by "OR," we combine them as follows:

\[
x < 10.33 \quad \text{OR} \quad x < 5.
\]

Simplifying, we note that any value satisfying \(x < 5\) will also satisfy \(x < 10.33\). Thus, the solution is:

\[
x < 10.33.
\]

However, since \(x < 5\) is a stricter condition, the final solution is:

\[
x < 10.33.
\]

Would you like more details or have any questions?

Here are some related questions for practice:

1. Solve: \(5x - 7 > 18 \, \text{OR} \, 2x + 3 < 7\).
2. Solve: \(-3x + 9 < 12 \, \text{OR} \, 7x - 5 > 20\).
3. Solve the inequality: \(4x - 15 \geq 3x + 2\).
4. Solve: \(-2x + 8 > 4 \, \text{OR} \, 3x - 6 < 9\).
5. Find the solution set for: \(6x - 11 < 13 \, \text{OR} \, -x + 4 > 5\).

**Tip:** When dealing with inequalities, always remember to flip the inequality sign when multiplying or dividing by a negative number!

Learn how to solve the compound inequality 3x - 8 < 23 OR -4x + 26 > 6 with a clear step-by-step approach. This solution covers key concepts like inequality manipulation and the rule for flipping inequalities when dividing by negative numbers. Ideal for students in Grades 7-9.

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