Answer :

To solve the inequality [tex]\(4x + 5 \geq 13\)[/tex], follow these steps:

1. Isolate the variable term on one side:
Start by subtracting 5 from both sides of the inequality:
[tex]\[ 4x + 5 - 5 \geq 13 - 5 \][/tex]
This simplifies to:
[tex]\[ 4x \geq 8 \][/tex]

2. Solve for [tex]\(x\)[/tex]:
Divide both sides of the inequality by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{4x}{4} \geq \frac{8}{4} \][/tex]
This simplifies to:
[tex]\[ x \geq 2 \][/tex]

Thus, the solution to the inequality is:
[tex]\[ x \geq 2 \][/tex]

3. Graph the solution:
To graph the solution [tex]\(x \geq 2\)[/tex]:
- Draw a number line.
- Identify the point [tex]\(x = 2\)[/tex] on the number line.
- Since the inequality includes "greater than or equal to" ([tex]\(\geq\)[/tex]), place a closed circle on [tex]\(x = 2\)[/tex] to indicate that 2 is included in the solution set.
- Shade the portion of the number line to the right of [tex]\(x = 2\)[/tex] to represent all values greater than 2.

The graph of the inequality [tex]\(4x + 5 \geq 13\)[/tex] looks like this:

[tex]\[ \begin{array}{c|ccccccccccccccc} \text{Number line:} & \cdots & 0 & 1 & \mathbf{2} & 3 & 4 & 5 & \cdots & \infty \\ \text{Graph:} & & & & \bullet & \rightarrow & \rightarrow & \rightarrow & \rightarrow & \cdots & \rightarrow \end{array} \][/tex]

In interval notation, this can be expressed as:
[tex]\[ [2, \infty) \][/tex]

Here, the closed interval bracket [ at 2 indicates that 2 is included in the solution set, and the infinity symbol indicates that the solution extends indefinitely to the right.