Solve the inequality and graph the solution.

8<3(v–2)–4
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.

1 answer

To solve the inequality \( 8 < 3(v - 2) - 4 \), we first simplify the right-hand side:

  1. Distribute the \( 3 \): \[ 3(v - 2) = 3v - 6 \] So the inequality becomes: \[ 8 < 3v - 6 - 4 \]

  2. Combine like terms on the right: \[ 3v - 6 - 4 = 3v - 10 \] Therefore, the inequality is: \[ 8 < 3v - 10 \]

  3. Add \( 10 \) to both sides to isolate the term with \( v \): \[ 8 + 10 < 3v \] This simplifies to: \[ 18 < 3v \]

  4. Divide both sides by \( 3 \) to solve for \( v \): \[ \frac{18}{3} < v \] Which simplifies to: \[ 6 < v \] or in standard form: \[ v > 6 \]

Graphing the Solution

  1. Plotting the endpoints: The solution is \( v > 6 \). We’ll plot the point \( 6 \).

  2. Closed vs. Open endpoints: Since the inequality is strictly greater than (not including \( 6 \)), we make the endpoint \( 6 \) open.

  3. Graphing the ray: The graph will show all values of \( v \) to the right of \( 6 \):

    • Use an open circle at \( 6 \) to indicate that \( 6 \) is not included in the solution.
    • Shade the region to the right of \( 6 \) to represent all numbers greater than \( 6 \).

Final Result

The solution to the inequality is \( v > 6 \), and the graph will look like this:

  • An open circle at \( 6 \).
  • A line extending to the right (toward positive infinity).

\[ \text{Graph:} \] \[ \text{(Open circle at 6, shading to the right)} \]