Question

Which graph below bests represent the solution set for `-2x+5+x\le3x+13`

Answer 1

To find the solution set for the inequality \(-2x + 5 + x \le 3x + 13\), we first simplify the inequality:

1. Combine like terms on the left-hand side:
\[
-2x + x + 5 \le 3x + 13
\]
This simplifies to:
\[
-x + 5 \le 3x + 13
\]

2. Next, we move all \(x\) terms to one side and constant terms to the other side. Adding \(x\) to both sides gives:
\[
5 \le 4x + 13
\]

3. Then, subtract \(13\) from both sides:
\[
5 - 13 \le 4x
\]
Simplifying this results in:
\[
-8 \le 4x
\]

4. To isolate \(x\), divide both sides by \(4\):
\[
-2 \le x
\]
or equivalently,
\[
x \ge -2
\]

The solution set is \(x \ge -2\).

### Graphical Representation

To represent this inequality graphically on a number line:

- The number \(-2\) will be marked on the number line.
- Since the inequality is \(x \ge -2\), we will use a closed dot (or filled circle) at \(-2\) to indicate that \(-2\) is included in the solution set.
- The region to the right of \(-2\) will be shaded to represent all the \(x\)-values greater than \(-2\).

Look for a graph that shows a closed dot at \(-2\) and shading to the right. This will best represent the solution set for the inequality \(x \ge -2\).