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.
