The inequality presented in the image is:

$-5 < 2x - 3 \leq 4$

Let's solve it step-by-step:

Step 1: Isolate $2x$

Start by adding 3 to all parts of the inequality:

$-5 + 3 < 2x - 3 + 3 \leq 4 + 3$

Simplifying this:

$-2 < 2x \leq 7$

Step 2: Solve for $x$

Now, divide the entire inequality by 2 to isolate $x$:

$\frac{-2}{2} < \frac{2x}{2} \leq \frac{7}{2}$

Simplifying:

$-1 < x \leq \frac{7}{2}$

So the solution is:

$-1 < x \leq \frac{7}{2}$

Interval Notation:

In interval notation, the solution is:

$(-1, \frac{7}{2}]$

This means $x$ is greater than $-1$ but less than or equal to $\frac{7}{2}$.

Graph:

• The number line would show an open circle at $x = -1$ (since $x$ does not include $-1$) and a closed circle at $x = \frac{7}{2}$ (since $x$ can equal $\frac{7}{2}$).
• The region between these two points would be shaded to represent the values of $x$.

Would you like a graph to be generated for this inequality?

Let me know if you want further clarifications or more details. Here are five related questions to explore:

1. How do we graph a compound inequality?
2. What is the difference between open and closed intervals on a number line?
3. Can you explain how interval notation works in inequalities?
4. What are the steps to solve compound inequalities like this one?
5. How does solving this inequality differ from solving an equation?

Tip:

Always ensure to perform operations on all parts of a compound inequality to maintain the relationship between the terms.