https://file.aiquickdraw.com/mathbot/file/1728526290510j4yzw98a.jpg

To write an inequality for Annette's earnings, let's break down the problem:

- Annette charges **$75** for each hair coloring, and **$45** for each haircut.
- She wants to earn at least **$450**.

Let:
- $ x $ be the number of hair colorings.
- $ y $ be the number of haircuts.

The total earnings from hair colorings and haircuts can be written as:

\[
75x + 45y
\]

Annette wants her earnings to be at least $450, so the inequality becomes:

\[
75x + 45y \geq 450
\]

Now, let's simplify this inequality by factoring out the greatest common factor of 15:

\[
15(5x + 3y) \geq 450
\]

Divide both sides by 15:

\[
5x + 3y \geq 30
\]

This is the simplified inequality representing the possible combinations of hair colorings and haircuts that Annette can do to earn at least $450.

Would you like further explanation or help with graphing this inequality? 

### 5 Related Questions:
1. How would you graph the inequality $ 5x + 3y \geq 30 $?
2. What is the significance of finding the intercepts of this inequality?
3. How does solving this inequality help Annette plan her work?
4. What is the meaning of the solution region on the graph?
5. How would Annette’s goal change if she wanted to earn at least $600 instead?

### Tip:
Always simplify inequalities by factoring out common terms, as it makes solving and graphing easier.

Annette works in a beauty salon. She charges $75 for hair coloring and $45 for haircuts. Annette would like to earn at least $450. Let x represent the number of hair colorings and y represent the number of haircuts. Write an inequality to represent the possible combinations.

Learn how to solve a linear inequality where Annette, a beautician, charges $75 for hair coloring and $45 for haircuts, and aims to earn at least $450. This problem demonstrates key concepts like linear inequalities, factoring, and graphing.

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