Solve the inequality and graph the solution.

Question

2.5m–1.5(2–3m)>2(m+1)
Plot the endpoints. Select an endpoint to change it from closed to open. Select the middle of the segment, ray, or line to delete it.

Answer 1

To solve the inequality \(2.5m - 1.5(2 - 3m) > 2(m + 1)\), we will start by simplifying both sides of the inequality.

1. Simplify the left side:

   \[ 2.5m - 1.5(2 - 3m) = 2.5m - 1.5 \times 2 + 1.5 \times 3m \] \[ = 2.5m - 3 + 4.5m \] \[ = (2.5m + 4.5m) - 3 \] \[ = 7m - 3 \]

2. Simplify the right side:

   \[ 2(m + 1) = 2m + 2 \]

3. Set the inequality:

   Now we combine both sides:

   \[ 7m - 3 > 2m + 2 \]

4. Subtract \(2m\) from both sides:

   \[ 7m - 2m - 3 > 2 \] \[ 5m - 3 > 2 \]

5. Add 3 to both sides:

   \[ 5m > 5 \]

6. Divide by 5:

   \[ m > 1 \]

Graphing the Solution

• The solution \(m > 1\) has an open endpoint at \(m = 1\) because the inequality does not include \(1\).
• Since it is a greater than inequality, we will shade to the right of \(1\).

Graph

• Draw a number line.
• Plot an open circle at point \(1\) (to indicate that \(1\) is not included).
• Shade the area to the right of \(1\) to indicate all numbers greater than \(1\).

Final Output:

The solution to the inequality is:

\[ m > 1 \]

And the graphical representation would look like this:

       |
     1 •--------------
       |

Where the circle at \(1\) is not filled (open circle) indicating that \(m\) can take any value greater than \(1\).