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The first song performed during a concert was more than 60 seconds longer than the final song. The first song was 190 seconds long. The inequality [tex]$190 > 60 + f$[/tex] represents the situation where [tex]$f$[/tex] is the length of the final song in seconds.

1. Which value should point [tex]$f$[/tex] be on the number line? [tex]\square[/tex]

2. Which best describes the appearance of the graph? [tex]\square[/tex]

Answer :

Sure! Let's solve the problem step-by-step:

### Step 1: Understand the given information
1. The first song was 190 seconds long.
2. The first song was more than 60 seconds longer than the final song. This can be written as:
[tex]\[
190 > 60 + f
\][/tex]
where [tex]\( f \)[/tex] is the length of the final song in seconds.

### Step 2: Simplify the inequality
We need to find [tex]\( f \)[/tex] such that the inequality holds true. Let's isolate [tex]\( f \)[/tex]:

1. Start with the given inequality:
[tex]\[
190 > 60 + f
\][/tex]
2. Subtract 60 from both sides to isolate [tex]\( f \)[/tex]:
[tex]\[
190 - 60 > f
\][/tex]
3. Simplify the left-hand side:
[tex]\[
130 > f
\][/tex]
or equivalently:
[tex]\[
f < 130
\][/tex]

This means that the length of the final song must be less than 130 seconds.

### Step 3: Identify the value and graph
1. Which value should point [tex]\( f \)[/tex] on the number line be? The value [tex]\( f \)[/tex] must be less than 130. This means [tex]\( f \)[/tex] can be any value shy of 130, such as 129, 100, 50, etc.

2. Which best describes the appearance of the graph? The graph of this inequality on a number line would be:

- A number line marked with values.
- An open circle (indicating that 130 is not included) at 130.
- A shaded line extending to the left of 130, representing all values less than 130.

So, we conclude:

- [tex]\( f \)[/tex] should be marked on the number line anywhere to the left of 130, not including 130 itself.
- The graph includes all numbers less than 130.

I hope this helps! Let me know if you have any questions.

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