Let's consider a real-world scenario to represent the inequality \(2x + 6 > 20\).
Scenario:
Rachel has [tex]$6 saved in her account, and she wants to have more than $[/tex]20 in total. She earns [tex]$2 for every hour she works. We want to find out how many hours Rachel needs to work in order to have more than $[/tex]20 in her account.
### Step-by-Step Solution:
1. Define the Variables:
- Let \(x\) represent the number of hours Rachel works.
- She earns $2 for each hour she works, so the total earnings from working \(x\) hours is \(2x\) dollars.
- Rachel already has $6 in her savings account.
2. Set Up the Inequality:
- The total amount of money Rachel will have after working \(x\) hours is the sum of her initial savings ([tex]$6) and her earnings from work ($[/tex]2x).
- This total should be more than $20.
Therefore, the inequality can be written as:
[tex]\[
2x + 6 > 20
\][/tex]
3. Solve the Inequality:
- First, we subtract 6 from both sides to isolate the term with \(x\):
[tex]\[
2x + 6 - 6 > 20 - 6
\][/tex]
Simplifies to:
[tex]\[
2x > 14
\][/tex]
- Next, we divide both sides by 2 to solve for \(x\):
[tex]\[
\frac{2x}{2} > \frac{14}{2}
\][/tex]
Simplifies to:
[tex]\[
x > 7
\][/tex]
4. Interpret the Solution:
- The solution \(x > 7\) means Rachel needs to work more than 7 hours in order to have more than $20 in her account.
Therefore, in this scenario, Rachel needs to work more than 7 hours to surpass the $20 mark in her savings.
