Solve the inequality: $12w + 45 > 7w + 85$.
The problem asks us to determine the condition on \(w\) such that the linear inequality \(12w + 45 > 7w + 85\) holds. This involves isolating the variable \(w\) to find the set of values that satisfy the condition. The inequality is a linear relationship between the expression \(12w + 45\) and \(7w + 85\). Our goal is to find when the first expression is greater than the second.
To isolate \(w\), we first collect all terms containing \(w\) on the left side by subtracting \(7w\) from both sides:
\[12w - 7w + 45 > 85\]
\[5w + 45 > 85\]
Next, we isolate the term with \(w\) by subtracting \(45\) from both sides of the inequality:
\[5w > 85 - 45\]
\[5w > 40\]
Finally, to solve for \(w\), we divide both sides by \(5\). Since \(5\) is a positive number, the direction of the inequality remains unchanged:
\[w > \frac{40}{5}\]
\[w > 8\]
This indicates that the inequality is true for any real number \(w\) that is greater than 8.
We have successfully determined that the inequality \(12w + 45 > 7w + 85\) is true if and only if \(w > 8\). For any value of \(w\) in the interval \((8, \infty)\), the condition is satisfied. The final answer is \(w > 8\).
Imagine you are comparing two phone plans:
Plan A costs \(\$45\) upfront plus \(\$12\) per month (\(12w + 45\)), and Plan B costs \(\$85\) upfront plus \(\$7\) per month (\(7w + 85\)). You want to know after how many months \(w\) Plan A becomes more expensive than Plan B. By solving the inequality \(12w + 45 > 7w + 85\), we find that Plan A costs more than Plan B after 8 months. This helps in budgeting for long-term services by finding the break-even point.