From the image, the handwritten problem appears to involve solving an inequality. Here's the transcription of the work and solution attempt from the notebook:

Problem:

$\frac{2x - 1}{x + 3} \leq -1$

Steps in the image:

1. Multiply both sides of the inequality by $x + 3$, assuming $x + 3 \neq 0$: $2x - 1 \leq -1(x + 3)$

2. Expand the right-hand side: $2x - 1 \leq -x - 3$

3. Add $x$ to both sides to combine like terms: $3x - 1 \leq -3$

4. Add 1 to both sides: $3x \leq -2$

5. Divide both sides by 3: $x \leq -\frac{2}{3}$

Now, to account for the domain restriction:

• The denominator $x + 3$ cannot be zero, so $x \neq -3$.
• Hence, the final solution for $x$ must also exclude this value.

Final solution:

$x \leq -\frac{2}{3}, \quad x \neq -3$

Would you like further clarification on this solution? Here are some related questions you might find helpful:

1. How does multiplying both sides of an inequality by a variable expression affect the inequality?
2. What are the domain restrictions in rational inequalities, and why are they important?
3. Can you provide more examples of solving rational inequalities?
4. What happens if we multiply or divide both sides of an inequality by a negative number?
5. How can you check if the solution to an inequality is correct?

Tip: Always remember to check the domain of the expression when solving rational inequalities to avoid including any undefined values.