2.2 – Applications of Linear Inequalities

Applications of Linear Inequalities

This section discusses solving problems using linear inequalities. The following steps provide a guideline to aid in solving such problems:

Solving Problems with Linear Inequalities
Read through the entire problem. Don’t worry about trying to solve anything, or about writing down what you might think is important yet. Focus on understanding what the problem is. Gather the important information. Look for information in the problem that is important, and discard information that isn’t. Look for key words to indicate the mathematical relations and operations. Create the mathematical model. Write the equation or inequality to describe the situation. Solve the mathematical model you have created. Check your answer. Plug your solution into the model and make sure it is a solution. Ask Yourself “Does my solution make sense?” We often find solutions to a problem that make little or no sense. We must ask ourselves if the solution we have actually makes sense for the problem.

Solving Problems with Linear Inequalities

Read through the entire problem. Don’t worry about trying to solve anything, or about writing down what you might think is important yet. Focus on understanding what the problem is.
Gather the important information. Look for information in the problem that is important, and discard information that isn’t. Look for key words to indicate the mathematical relations and operations.
Create the mathematical model. Write the equation or inequality to describe the situation.
Solve the mathematical model you have created.
Check your answer. Plug your solution into the model and make sure it is a solution.
Ask Yourself “Does my solution make sense?” We often find solutions to a problem that make little or no sense. We must ask ourselves if the solution we have actually makes sense for the problem.

When looking for the key words (in step 2) it is helpful to keep the following words and their meanings in mind:

$\begin{align*} & & \text{less than} & \ \ \ \ < & \text{at least} & \ \ \ \ \ge & & \\ & & \text{more than} & \ \ \ \ > & \text{at most} & \ \ \ \ \le \\ & & \text{fewer than} & \ \ \ \ < & \text{minimum} & \ \ \ \ \ge \\ & & \text{greater than} & \ \ \ \ > & \text{maximum} & \ \ \ \ \le \\ \end{align*}$

We can see this in the following examples:

Jonathan works as a salesman earning a base wage of $30,000 plus a commission of 5% of his sales. If he sets a personal goal to earn at least $42,500 in a year, what will his total sales need to be?
The important information here is that he has a base salary of $30,000, a commission of 5%, and wants to earn at least $42,500. We can create our equation using $x$ to represent the total sales Jonathan must make: $30000+0.05x\ge42500$ And we solve by first subtracting 30000 from each side $0.05x\ge12500$ and then dividing each side by 0.05 $x\ge250000$ So Jonathan needs sales of at least $250,000.

Jonathan works as a salesman earning a base wage of $30,000 plus a commission of 5% of his sales. If he sets a personal goal to earn at least $42,500 in a year, what will his total sales need to be?

The important information here is that he has a base salary of $30,000, a commission of 5%, and wants to earn at least $42,500. We can create our equation using $x$ to represent the total sales Jonathan must make:

$30000+0.05x\ge42500$
And we solve by first subtracting 30000 from each side
$0.05x\ge12500$
and then dividing each side by 0.05
$x\ge250000$

So Jonathan needs sales of at least $250,000.

A taxi cab charges a fixed amount of $5.75 plus a charge of $1.25 per mile. If Heather only has $40, what is the furthest she can take the taxi?
Note that in the problem, we are limited in the amount of money we can spend. The taxi fare has to be less than or equal to $40. We can use this information to set up the problem using $x$ as the number of miles traveled: $5.75+1.25x\le40$ and we start to solve by subtracting 5.75 from each side: $1.25x\le34.25$ and divide each side by 1.25: $x\le 27.4$ So Heather can take the taxi 27.4 miles or less

A taxi cab charges a fixed amount of $5.75 plus a charge of $1.25 per mile. If Heather only has $40, what is the furthest she can take the taxi?

Note that in the problem, we are limited in the amount of money we can spend. The taxi fare has to be less than or equal to $40. We can use this information to set up the problem using $x$ as the number of miles traveled:

$5.75+1.25x\le40$
and we start to solve by subtracting 5.75 from each side:
$1.25x\le34.25$
and divide each side by 1.25:
$x\le 27.4$

So Heather can take the taxi 27.4 miles or less

Andy goes to the store to purchase drinks for his party. He also purchases a magazine for $5. If each drink costs $3 and he he brought $47 with him, what is the maximum number of drinks he can buy?
We can set up the following inequality: $5+3x\le47$ And solve: $3x\le42$ $x\le14$ So Andy can buy 14 drinks or less.

Jim needs an average of at least 80 on his tests in his math course in order to earn a “B” grade, If his scores on the first three tests are 78, 84 and 86, what is the minimum he can score on his fourth test?
Jim’s average of all four tests must be greater than 80. The average of four tests is their sum divided by the number of tests (4 total) so we have: $\frac{78+84+86+x}{4}\ge80$ Where $x$ is the unknown fourth test score. We will begin to solve this inequality by multiplying both sides by 4. This will eliminate the fraction on the left: ${\color{red}4\cdot}\frac{78+84+86+x}{4}\ge{\color{red}4\cdot}80$$ $78+84+86+x\ge320$ We can then add like terms on the left: $248+x\ge320$ And then subtract 248 from each side: $x\ge72$

Jim needs an average of at least 80 on his tests in his math course in order to earn a “B” grade, If his scores on the first three tests are 78, 84 and 86, what is the minimum he can score on his fourth test?

Jim’s average of all four tests must be greater than 80. The average of four tests is their sum divided by the number of tests (4 total) so we have:
$\frac{78+84+86+x}{4}\ge80$
Where $x$ is the unknown fourth test score. We will begin to solve this inequality by multiplying both sides by 4. This will eliminate the fraction on the left:
${\color{red}4\cdot}\frac{78+84+86+x}{4}\ge{\color{red}4\cdot}80$$
$78+84+86+x\ge320$
We can then add like terms on the left:
$248+x\ge320$
And then subtract 248 from each side:
$x\ge72$

PreCalculus I

2.2 – Applications of Linear Inequalities

Applications of Linear Inequalities

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