To plot the solutions to the inequality [tex]\(\frac{x}{2} > \frac{5}{2}\)[/tex] on a number line, let's go through the steps in detail:
1. Rewrite the Inequality in a Simpler Form:
The given inequality is [tex]\(\frac{x}{2} > \frac{5}{2}\)[/tex].
- To eliminate the fractions, multiply both sides of the inequality by 2.
[tex]\[
2 \cdot \frac{x}{2} > 2 \cdot \frac{5}{2}
\][/tex]
- This simplifies to:
[tex]\[
x > 5
\][/tex]
2. Identify the Solution Set:
The inequality [tex]\(x > 5\)[/tex] means that [tex]\(x\)[/tex] can be any number greater than 5.
3. Determine the Type of Endpoint:
- The inequality is strict (i.e., [tex]\(x\)[/tex] is not equal to 5, but greater than 5).
- This means we will use an open circle to show that 5 is not included in the solution set.
4. Plot on a Number Line:
- Draw a horizontal line to represent the number line.
- Mark the point [tex]\(5\)[/tex] on the number line.
- Place an open circle at [tex]\(5\)[/tex] to indicate that [tex]\(5\)[/tex] is not included.
- Shade the region to the right of [tex]\(5\)[/tex] to show all numbers greater than [tex]\(5\)[/tex].
Here is how the number line looks:
[tex]\[
\begin{array}{c|cccccccccccccccccc}
\text{Number line:} &&& & \circ & \longrightarrow & & & & & & & & & & \\
&&& & 5 & & & & & & & & & & & & & & \\
\end{array}
\][/tex]
- The open circle at [tex]\(5\)[/tex] indicates that [tex]\(5\)[/tex] itself is not part of the solution.
- The shading to the right of [tex]\(5\)[/tex] shows that all numbers greater than [tex]\(5\)[/tex] are included in the solution set.
