1) On the number line, distance from 5x+15 to origin must be greater than or equal to 25. a. Frame the inequality. b. Graph it and solve for the solution set.

### Part (a): Framing the Inequality

To frame the inequality, we consider the distance from \( 5x + 15 \) to the origin. The distance on the number line from a point \( a \) to the origin is given by \( |a| \). In this case, the distance is \( |5x + 15| \).

The problem states that this distance must be greater than or equal to 25. Therefore, the inequality can be framed as:

\[
|5x + 15| \geq 25
\]

### Part (b): Solving the Inequality

To solve the inequality \( |5x + 15| \geq 25 \), we consider two cases:

#### Case 1: 
\[
5x + 15 \geq 25
\]
Subtract 15 from both sides:
\[
5x \geq 10
\]
Divide by 5:
\[
x \geq 2
\]

#### Case 2: 
\[
5x + 15 \leq -25
\]
Subtract 15 from both sides:
\[
5x \leq -40
\]
Divide by 5:
\[
x \leq -8
\]

### Solution Set:
The solution set combines both cases:

\[
x \leq -8 \quad \text{or} \quad x \geq 2
\]

### Graphing the Inequality
The graph on the number line will show the intervals where \( x \leq -8 \) or \( x \geq 2 \).

1. Mark the point \( -8 \) and \( 2 \) on the number line.
2. Shade the region to the left of \( -8 \) (including \( -8 \)) and to the right of \( 2 \) (including \( 2 \)).

Let me know if you would like the graphical representation, a detailed explanation, or if you have any questions!

### Related Questions:

1. How do you solve inequalities involving absolute values?
2. Can you graph the inequality \( |x - 3| \leq 7 \)?
3. How does the solution of \( |5x + 15| < 25 \) differ from \( |5x + 15| \geq 25 \)?
4. What is the significance of the origin in distance inequalities?
5. How do you interpret inequalities graphically on a number line?
6. Can you solve and graph \( |2x - 4| > 10 \)?
7. How do you find the distance between two points on a number line?
8. How does inequality change if the inequality sign is reversed?

**Tip:** When solving absolute value inequalities, always consider both the positive and negative cases.

Learn how to solve the absolute value inequality |5x + 15| >= 25, determining the distance from 5x + 15 to the origin on a number line. Includes step-by-step solutions, graphical representation, and related concepts. Suitable for high school students.

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