To solve the inequality \( 8 < 3(v - 2) - 4 \), we first simplify the right-hand side:
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Distribute the \( 3 \): \[ 3(v - 2) = 3v - 6 \] So the inequality becomes: \[ 8 < 3v - 6 - 4 \]
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Combine like terms on the right: \[ 3v - 6 - 4 = 3v - 10 \] Therefore, the inequality is: \[ 8 < 3v - 10 \]
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Add \( 10 \) to both sides to isolate the term with \( v \): \[ 8 + 10 < 3v \] This simplifies to: \[ 18 < 3v \]
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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
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Plotting the endpoints: The solution is \( v > 6 \). We’ll plot the point \( 6 \).
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Closed vs. Open endpoints: Since the inequality is strictly greater than (not including \( 6 \)), we make the endpoint \( 6 \) open.
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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).
\[ \text{Graph:} \] \[ \text{(Open circle at 6, shading to the right)} \]
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