Writing and solving inequalities can help us make sense of the constraints in a situation and solve problems. Let's look at an example.

Clare would like to buy a video game system that costs $130 and to have some extra money for games. She has saved $48 so far and plans on saving $5 of her allowance each week. How many weeks, $w$, will it be until she has enough money to buy the system and have some extra money remaining? To represent the constraints, we can write $48 + 5w \geq 130$. Let’s reason about the solutions:

• Because Clare has $48 already and needs to have at least $130 to afford the game system, she needs to save at least $82 more.
• If she saves $5 each week, it will take at least LATEX2 weeks to reach $82.
• $\frac{82}{5}$ is 16.4. Any time shorter than 16.4 weeks won't allow her to save enough.
• Assuming she saves $5 at the end of each week (instead of saving smaller amounts throughout a week), it will be at least 17 weeks before she can afford the game system. 

We can also solve by writing and solving a related equation to find the boundary value for $w$, and then determine whether the solutions are less than or greater than that value.

Sometimes the structure of an inequality can help us see whether the solutions are less than or greater than a boundary value. For example, to find the solutions to $3x > 8x$, we can solve the equation $3x = 8x$, which gives us $x = 0$. Then, instead of testing values on either side of 0, we could reason as follows about the inequality:

• If $x$ is a positive value, then $3x$ would be less than $8x$.
• For $3x$ to be greater than $8x$, $x$ must include negative values.
• For the solutions to include negative values, they must be less than 0, so the solution set would be $x < 0$.