https://file.aiquickdraw.com/mathbot/file/1726715456437inecfcfg.jpg

The problem in your image is asking to solve the compound inequality:

\[
5x - 4 \leq -3 \quad \text{or} \quad 4x + 6 > 0
\]

and then write the solution in interval notation. Let's solve each part step by step.

### Solving the first inequality: \( 5x - 4 \leq -3 \)
1. Add 4 to both sides:
   \[
   5x \leq -3 + 4
   \]
   \[
   5x \leq 1
   \]
2. Divide both sides by 5:
   \[
   x \leq \frac{1}{5}
   \]

### Solving the second inequality: \( 4x + 6 > 0 \)
1. Subtract 6 from both sides:
   \[
   4x > -6
   \]
2. Divide both sides by 4:
   \[
   x > -\frac{6}{4}
   \]
   Simplify the fraction:
   \[
   x > -\frac{3}{2}
   \]

### Combining the solutions
The solution to the compound inequality is the union of the two individual solutions:
\[
x \leq \frac{1}{5} \quad \text{or} \quad x > -\frac{3}{2}
\]

### Interval notation
- For \( x \leq \frac{1}{5} \), the interval is \( (-\infty, \frac{1}{5}] \).
- For \( x > -\frac{3}{2} \), the interval is \( (-\frac{3}{2}, \infty) \).

Since this is an "or" inequality, we take the union of these two intervals. The union of \( (-\infty, \frac{1}{5}] \) and \( (-\frac{3}{2}, \infty) \) is:
\[
(-\infty, \infty)
\]
Thus, the solution is all real numbers, written as \( (-\infty, \infty) \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How would the solution change if the inequalities were connected with "and" instead of "or"?
2. Can you graph the solution to this compound inequality?
3. What is the difference between solving "and" and "or" compound inequalities?
4. How can interval notation be used to express the solution to inequalities involving absolute values?
5. How do you solve inequalities involving fractions?

**Tip:** Always check the direction of inequalities when multiplying or dividing by negative numbers; you need to reverse the inequality sign.

Solve 5x - 4 ≤ -3 or 4x + 6 > 0 and write the solution in interval notation.

Learn how to solve compound inequalities like 5x - 4 ≤ -3 or 4x + 6 > 0. This problem covers solving inequalities and expressing solutions using interval notation, ideal for students in Grades 8-10.

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