# Equations and Inequalities

## Models and Applications


### Learning Objectives

1. Solve a formula for a specified variable (IA 2.3.1)
2. Use a problem-solving strategy for word problems (IA 2.2.1)

### Objective 1: Solve a formula for a specified variable (IA 2.3.1)


### Practice Makes Perfect

Solve each formula for the specific variable.

### Objective 2: Use a problem-solving strategy for word problems (IA 2.2.1)


### Practice Makes Perfect

Use a Problem-Solving Strategy for word problems.
Neka is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only the final exam remains, and the maximum number of points that can be earned is 100. Is it possible for Neka to end the course with an A? A simple linear equation will give Neka his answer.
Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

### Setting up a Linear Equation to Solve a Real-World Application

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write 
  This expression represents a variable cost because it changes according to the number of miles driven.
If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. We can use these quantities to model an equation that can be used to find the daily car rental cost 
  
When dealing with real-world applications, there are certain expressions that we can translate directly into math.  lists some common verbal expressions and their equivalent mathematical expressions.

### Using a Formula to Solve a Real-World Application

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region, 
  the perimeter of a rectangle, 
  and the volume of a rectangular solid, 
  When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

### Key Concepts

1. A linear equation can be used to solve for an unknown in a number problem. See .
2. Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. See .
3. There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the 
  formula. See .
4. Many geometry problems are solved using the perimeter formula 
  the area formula 
  or the volume formula 
  See , , and .

### Section Exercises


### Verbal


### Real-World Applications

For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked.
For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges of $.05/min for calls. Company B has a monthly fee of $5 and charges $.10/min for calls.
For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use 
  for the number of devices that need data plans as part of their cost.
For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.
For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus $.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.
For the following exercises, use the formula given to solve for the required value.
For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.