https://file.aiquickdraw.com/mathbot/file/1727879617135b6cram3j.jpg

Let's solve the given inequality and analyze the graph choices. 

### Given Problem:
Solve the linear inequality and then:
- Graph the solution on a number line.
- Write the solution in set-builder notation.
- Write the solution in interval notation.

The inequality is:

\[
5(t - 2) - 7t \geq t + 11
\]

### Step 1: Simplify the Inequality
1. Expand \(5(t - 2)\):
   \[
   5t - 10 - 7t \geq t + 11
   \]

2. Combine like terms involving \(t\):
   \[
   (5t - 7t) - 10 \geq t + 11
   \]
   \[
   -2t - 10 \geq t + 11
   \]

3. Add \(2t\) to both sides:
   \[
   -10 \geq 3t + 11
   \]

4. Subtract 11 from both sides:
   \[
   -10 - 11 \geq 3t
   \]
   \[
   -21 \geq 3t
   \]

5. Divide both sides by 3:
   \[
   -7 \geq t
   \]
   This can be rewritten as:
   \[
   t \leq -7
   \]

### Step 2: Graph the Solution
The solution is \( t \leq -7 \). We need to represent this inequality on a number line.

- Since \(t \leq -7\), the solution includes all values less than or equal to -7.
- On the graph, this would be a filled circle at \(-7\) (to indicate inclusion) and an arrow extending to the left.

From the provided graphs:
- **Graph B** represents \( t \leq -7 \) with a filled circle at -7 and an arrow pointing to the left.

Thus, the correct graph is **B**.

### Step 3: Write the Solution in Set-Builder Notation
In set-builder notation, the solution is:

\[
\{ t \mid t \leq -7 \}
\]

### Step 4: Write the Solution in Interval Notation
In interval notation, the solution is:

\[
(-\infty, -7]
\]

### Summary
- **Graph**: B
- **Set-Builder Notation**: \(\{ t \mid t \leq -7 \}\)
- **Interval Notation**: \((-\infty, -7]\)

Would you like more details on any of these steps? 

Here are 5 related questions that might expand on this topic:
1. How do you determine which type of circle (filled or open) to use in graphing inequalities?
2. What are some examples of inequalities that require similar steps to solve?
3. How does solving an inequality differ from solving an equation?
4. How would the solution change if the inequality were \(5(t - 2) - 7t > t + 11\)?
5. How can interval notation be used to represent unions of multiple intervals?

**Tip**: Always remember to reverse the inequality sign when multiplying or dividing by a negative number during your steps.

Solve the linear inequality 5(t - 2) - 7t ≥ t + 11, graph the solution on a number line, and write the solution in set-builder and interval notation.

Learn how to solve the linear inequality 5(t - 2) - 7t ≥ t + 11 step by step. Graph the solution on a number line, and understand how to express the solution in both set-builder and interval notation. Perfect for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation