To solve the inequality [tex]\(\frac{1}{4}x < \frac{1}{2}\)[/tex], we need to isolate [tex]\(x\)[/tex].
Step-by-step process:
1. Start with the given inequality:
[tex]\[
\frac{1}{4} x < \frac{1}{2}
\][/tex]
2. To isolate [tex]\(x\)[/tex], we need to eliminate the fraction. Multiply both sides of the inequality by 4 (which is the reciprocal of [tex]\(\frac{1}{4}\)[/tex]):
[tex]\[
\left( \frac{1}{4} x \right) \cdot 4 < \left( \frac{1}{2} \right) \cdot 4
\][/tex]
3. Simplify both sides:
[tex]\[
x < 2
\][/tex]
Thus, the solution to the inequality is [tex]\(x < 2\)[/tex].
To represent this on a number line:
1. Draw a horizontal line to represent the number line.
2. Locate the point 2 on the number line.
3. Draw an open circle at 2 to indicate that 2 is not included in the solution set.
4. Shade the region to the left of 2 to indicate all numbers less than 2.
Here is the visual representation:
[tex]\[
\begin{array}{c}
\text{Number Line:} \\
\begin{tikzpicture}
\draw[<->] (-4,0) -- (4,0);
\foreach \x in {-3,-2,-1,0,1,2,3}
\draw (\x,0.1) -- (\x,-0.1) node[below] {\footnotesize $\x$};
\draw[fill=white] (2,0) circle (0.1cm);
\draw[-] (-4,0) -- (2,0);
\end{tikzpicture}
\end{array}
\][/tex]
This visual representation shows all [tex]\(x\)[/tex] values less than 2.
