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

Note: Use improper fractions, if necessary.

We have the following compound inequality to solve:

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

### Step 1: Solve the first inequality \( 5x - 4 \leq -3 \)

Add 4 to both sides:

\[
5x \leq -3 + 4
\]
\[
5x \leq 1
\]

Now, divide both sides by 5:

\[
x \leq \frac{1}{5}
\]

### Step 2: Solve the second inequality \( 4x + 6 > 0 \)

Subtract 6 from both sides:

\[
4x > -6
\]

Now, divide both sides by 4:

\[
x > -\frac{6}{4}
\]
Simplifying the fraction:

\[
x > -\frac{3}{2}
\]

### Step 3: Combine the results

We need to combine the solutions from both inequalities. The final solution is:

\[
x \leq \frac{1}{5} \quad \text{or} \quad x > -\frac{3}{2}
\]

### Step 4: Write the solution in interval notation

The solution in interval notation is:

\[
(-\infty, \frac{1}{5}] \cup \left(-\frac{3}{2}, \infty \right)
\]

This is the final solution.

---

Would you like further details on any step? 

Here are some related questions to practice:
1. How do you solve compound inequalities involving "and" and "or"?
2. What is the interval notation for solutions involving infinity?
3. How would the solution change if one of the inequalities was strict (e.g., \( < \) or \( > \))?
4. Can compound inequalities have no solutions?
5. How can you graph the solution on a number line?

**Tip:** When solving inequalities, always remember to reverse the inequality sign if multiplying or dividing by a negative number.

Solve 5x - 4 ≤ -3 or 4x + 6 > 0 and write the solution in interval notation. Note: Use improper fractions, if necessary.

Learn how to solve the compound inequality 5x - 4 ≤ -3 or 4x + 6 > 0 and express the solution in interval notation. This step-by-step guide covers key concepts in algebra, including inequalities and interval notation. Ideal for students in Grades 9-11.

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