• Solve applications with linear inequalities

Assignment

Solve Applications with Linear Inequalities

These problems require the same strategies as other word problems, save for now we need to set up inequalities instead of equations. Making sure your answers make sense are important. You should be writing short sentences describing the answer, this way you can hopefully catch something that doesn’t make sense.

Example 1

Emma got a new job and will have to move. Her monthly income will be $5,625. To qualify to rent an apartment, Emma’s monthly income must be at least three times as much as the rent. What is the highest rent Emma will qualify for?

There is one part we need to focus on here: “Emma’s monthly income must be at least three times as much as the rent”.

We know her income, and three times the unknown rent can be $3r$. The “at least” means we have to use an inequality symbol, and getting it to point in the right direction is important.

Since income is tied to the words “at least”, we know a higher rent is OK. Lower is not.

\[\begin{align} 5625 \ge 3r \\ 1875 \ge r \end{align}\]

Or you can reverse it so that it’s easier to read.

\[\begin{align} r \le 1875 \end{align}\]

Emma qualifies for rent of $1875 or lower.

$\blacksquare$

Example 2

Dawn won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?

The cost of the tablets can be figured out by multiplying how many are bought by the cost. And whatever that comes out to be, that has to be covered by the $4000 grant.

\[\begin{align} 254.12n \le 4000 \\ n \le 15.74 \end{align}\]

OK, but that’s not a good answer. You can’t buy $0.74$ of a tablet. Dawn can buy up to $15$ tablets for her classroom.

$\blacksquare$

Example 3

Pete works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925?

Any of you that had to go through the portfolio process might recognize this kind of problem. We want the sales-based salary to beat the fixed one, but we need a way to express the sales-based salary.

Well, he will make $500 no matter what, but also 12% of his sales, so $0.12s$. Since we want this to beat the other salary, we’ll use greater than without the equals.

\[\begin{align} 500 + 0.12s > 925 \\ 0.12s > 425 \\ s > 3541.67 \end{align}\]

In order to beat the other salary, he needs to make sales totaling more than $3541.67.

$\blacksquare$