Let me analyze the questions in the image and provide solutions.

Question 6:

Solve the following equations:

(1) $13x + 5 = 0$

• Rearranging: $13x = -5$ $x = \frac{-5}{13}$ Therefore, $x = -\frac{5}{13}$.

(2) $2x + 1 = 5$

• Rearranging: $2x = 5 - 1$ $2x = 4$ $x = \frac{4}{2}$ So, $x = 2$.

Question 7:

Solve the following inequalities:

(1) $|2x - 3| \geq 5$

This is an absolute value inequality. We split it into two cases:

1. $2x - 3 \geq 5$ $2x \geq 8$ $x \geq 4$

2. $2x - 3 \leq -5$ $2x \leq -2$ $x \leq -1$

The solution is $x \leq -1$ or $x \geq 4$.

(2) $1 \leq |2x + 1| < 5$

We split the absolute value into two inequalities and solve them separately:

For $|2x + 1| \geq 1$:

1. $2x + 1 \geq 1$ $2x \geq 0$ $x \geq 0$

2. $2x + 1 \leq -1$ $2x \leq -2$ $x \leq -1$

For $|2x + 1| < 5$: $-5 < 2x + 1 < 5$ Subtract 1 from all sides: $-6 < 2x < 4$ Divide by 2: $-3 < x < 2$

Thus, the solution for $1 \leq |2x + 1| < 5$ is $x \in [-3, -1] \cup [0, 2)$.

Question 8:

Given that $x$ is an integer and $|2x - 1| \leq x + 3$, find how many solutions there are.

We need to solve $|2x - 1| \leq x + 3$, which can be split into two cases:

1. $2x - 1 \leq x + 3$ $2x - x \leq 3 + 1$ $x \leq 4$

2. $-(2x - 1) \leq x + 3$ (which simplifies to $-2x + 1 \leq x + 3$): $-2x - x \leq 3 - 1$ $-3x \leq 2$ $x \geq -\frac{2}{3}$

Since $x$ is an integer, the valid solutions for $x$ are $-1, 0, 1, 2, 3, 4$.

Thus, there are 6 solutions.

Do you have any questions or would you like more details on any part of the solution?

Here are 5 related questions to expand on this topic:

1. How do you solve inequalities involving absolute values?
2. What is the difference between "greater than" and "greater than or equal to" in inequalities?
3. How can we graph solutions to inequalities on a number line?
4. What are the properties of absolute value equations?
5. How can inequalities be applied in real-world problems?

Tip: When solving absolute value inequalities, always consider both positive and negative cases, as the absolute value function creates two different scenarios.