Equations and Inequalities

Models and Applications

Mathematical models use equations to represent real-world situations, allowing us to solve practical problems by translating words into algebraic expressions. The process of solving word problems with linear equations involves a systematic approach to ensure accuracy and clarity.

• Step 1: Let x represent one of the unknown quantities.

• Step 2: Represent other unknown quantities in terms of x.

• Step 3: Write an equation in x that models the conditions of the problem.

• Step 4: Solve the equation and answer the question.

• Step 5: Check the solution in the original wording of the problem, not just in the equation.

Example 1: Salary Comparison Problem

This example demonstrates how to use a linear equation to compare the median starting salaries of three majors, given their relationships and total sum.

• Let x: Median starting salary of an education major.

• Computer Science: x + 21 (thousand dollars)

• Economics: x + 14 (thousand dollars)

• Equation: x + (x + 21) + (x + 14) = 140

• Solution: x = 35 (Education), 56 (Computer Science), 49 (Economics)

• Verification: 35 + 56 + 49 = 140

Example 2: Years Spent Sleeping and Eating

This problem asks you to determine the number of years spent on two activities, given their sum and the difference between them.

• Let x: Years spent eating.

• Years sleeping: x + 24

• Equation: x + (x + 24) = 32

• Solution: 2x + 24 = 32 \Rightarrow 2x = 8 \Rightarrow x = 4

• Years eating: 4

• Years sleeping: 28

Example 3: Comparing Texting Plans

Linear equations can be used to compare costs and determine break-even points between two pricing plans.

• Let x: Number of text messages.

• Plan A:

• Plan B:

• Equation:

• Solution:

• Verification: Both plans cost $39 for 300 messages.

Example 4: Bridge Toll Cost Comparison

This example involves finding the number of times a bridge must be crossed for two payment options to cost the same.

• Let x: Number of bridge crossings per month.

• Without pass:

• With pass:

• Equation:

• Solution:

• Verification: For 20 crossings, both options cost $100.

Example 5: Price Reduction Problem

Solving for the original price before a percentage reduction is a common application of linear equations.

• Let x: Price before reduction.

• Equation:

• Solution:

• Verification:

Example 6: Investment Problem

Problems involving investments at different interest rates can be modeled with linear equations to find the amount invested at each rate.

• Let x: Amount invested at 9%.

• Amount at 11%:

• Equation:

• Solution: at 9%, $1850$ at 11%

• Verification:

Example 7: Basketball Court Dimensions

Geometry problems often require setting up equations based on perimeter or area formulas.

• Let x: Width of the court.

• Length:

• Equation:

• Solution: ft (width), ft (length)

• Verification:

Solving a Formula for a Variable

Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side of the equation. This is useful for rearranging equations to solve for different quantities in applications.

• Example: Rearranging the area formula for a rectangle to solve for gives .

Common Formulas for Area, Perimeter, and Volume

Many application problems require knowledge of standard geometric formulas. The following table summarizes common formulas used in College Algebra.

Additional info: The examples and table above provide a foundation for solving a wide range of application problems in College Algebra, including those involving geometry, finance, and everyday decision-making.