To solve the inequality:
[tex]\[
4x \geq 32
\][/tex]
we need to isolate the variable [tex]\( x \)[/tex] by performing operations that will keep the inequality true.
1. Divide both sides of the inequality by 4:
[tex]\[
\frac{4x}{4} \geq \frac{32}{4}
\][/tex]
This simplifies to:
[tex]\[
x \geq 8
\][/tex]
The solution to the inequality [tex]\( x \geq 8 \)[/tex] implies that [tex]\( x \)[/tex] is any number greater than or equal to 8.
2. Expressing the solution set in interval notation:
In interval notation, the solution set is written as:
[tex]\[
[8, \infty)
\][/tex]
This notation indicates that [tex]\( x \)[/tex] includes 8 and extends to positive infinity.
3. Graphing the solution set:
To represent this graphically:
- Draw a number line.
- Place a closed dot or bracket at [tex]\( x = 8 \)[/tex] to indicate that 8 is included in the solution (since the inequality is "greater than or equal to").
- Shade the number line to the right of 8 to indicate all numbers greater than 8.
Here's a representation of the graph:
[tex]\[
\begin{array}{lcl}
\text{---} & \cdots & \text{---} \hspace{-5pt}\bullet \hspace{-10pt}\text{======}\hspace{3pt}\rightarrow \\
\text{-10} & & \text{8} \\
\end{array}
\][/tex]
The solution set in interval notation is:
[tex]\[
[8, \infty)
\][/tex]
