Question

Solve the inequality and graph the solution.
8 ≥ t+ 9
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. Solve the inequality and graph the solution.
8 ≥ t+ 9
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.

Answers

Answered by GPT-4o mini
To solve the inequality \( 8 \geq t + 9 \), we will isolate \( t \):

1. Subtract \( 9 \) from both sides:
\[
8 - 9 \geq t
\]
\[
-1 \geq t
\]
or equivalently,
\[
t \leq -1
\]

This tells us that \( t \) can be any value that is less than or equal to \(-1\).

**Graphing the solution:**

1. The solution set is \( (-\infty, -1] \).
2. Since \(-1\) is included in the solution (because \( t \) can be equal to \(-1\)), we will represent this with a closed endpoint at \(-1\).
3. The graph will extend to the left (indicating all values less than \(-1\)).

**Steps to Graph:**
- Draw a number line.
- Mark the point \(-1\).
- Shade the line to the left of \(-1\) to indicate all values less than \(-1\).
- Put a closed dot (●) at \(-1\) to show that it is included in the inequality.

**Changing an Endpoint to Open:**
If we need to change an endpoint to open, we can simply say we will change the \( -1 \) into an open dot if we consider \( t < -1 \) which is not part of our original inequality.

**Deleting Middle of the Segment:**
If we interpret deleting the middle of the segment as potentially altering the way we see the inequality, we would outline the values less than \(-1\) without including \(-1\) itself.

So let’s summarize:
- The original solution is \( t \leq -1 \), which is graphed as a closed endpoint at \(-1\) with a shaded line extending to the left.
- If one endpoint (like \(-1\)) is made open, we could say \( t < -1 \) with an open dot at \(-1\) and shading to the left.

To clarify, we deal with the original inequality, and the method of deleting or adjusting endpoints is based on the context of the problem or the operational changes made to the inequality.

Here is a simple description of the graph:
- A number line with an open dot at \(-1\) if we change the inequality to less than, or closed dot if we keep it as \( \leq -1 \).
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