Problem 132 - Algebra and Trigonometry

Question

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. Parts for an automobile repair cost 175 dollar. The mechanic charges 34 dollar per hour. If you receive an estimate for at least 226 dollar and at most 294 dollar for fixing the car, what is the time interval that the mechanic will be working on the job?

Step-by-Step Solution

Answer

The mechanic will work anywhere from 1.5 hours to 3.5 hours.

Step 1: Formulate the inequality

The total cost is made up of the fixed cost of the parts, and the mechanic's charge per hour. Therefore, the equation can be written as \(175 + 34x \leq 294\) and \(175 + 34x \geq 226\), where \(x\) is the number of hours the mechanic will work.

Step 2: Solve the inequalities

Solving the first inequality: \(175 + 34x \leq 294\), we subtract 175 from both sides to get \(34x \leq 119\). Then, we divide by 34 to get \(x \geq 3.5\) hours. For the second inequality: \(175 + 34x \geq 226\), we subtract 175 from both sides to get \(34x \geq 51\). Finally, dividing by 34 gives \(x \leq 1.5\) hours.

Step 3: Interpret the solution

The solution \(x \geq 1.5\) and \(x \leq 3.5\) represents the interval of time the mechanic will be working on the job. This means that the mechanic will work anywhere from 1.5 hours to 3.5 hours.

Key Concepts

Word Problems

When faced with word problems in algebra, the key is to transform the given verbal conditions into mathematical statements. The first step always involves understanding the context and the question being asked. In our example, we need to find the time interval that a mechanic will work on a car, given a cost estimate range.

It's essential to identify the variables and constants. Here, the variable is the number of hours the mechanic will work (represented by 'x') and the constants are the parts cost and the mechanic's hourly rate. Critical thinking is required to decide what operations to use—addition, subtraction, multiplication, or division—to model the situation. Practice translating sentences of the problem into equations or inequalities is invaluable for mastering word problems.

Inequality Equations

Inequality equations set the stage for comparing different mathematical expressions. Unlike equations, which state that two quantities are equal, inequalities tell us how one quantity is less than, greater than, less than or equal to, or greater than or equal to another quantity. Dealing with inequalities, like the one in our exercise, requires similar steps to solving equations: isolate the variable to one side while keeping the inequality balanced.

To solve inequality equations, manipulation involves adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. Here we must remember that multiplying or dividing by a negative number reverses the inequality sign. Being proficient in these techniques is critical to finding the solution to inequality equations.

Algebraic Modeling

Algebraic modeling is about representing real-world scenarios with algebraic expressions or equations. The goal is to create a model that we can manipulate to predict outcomes or understand relationships within the given context. When modeling a situation, it's vital to identify which elements vary and which remain constant.

In our car repair scenario, the total cost of the repair job varies depending on the hours worked, so we represent this with a linear inequality. The term linear indicates that the hours and total cost have a direct proportional relationship, as seen in the constant rate of the mechanic's hourly charge. By using algebraic modeling, we can make abstract relationships tangible and solvable, enabling us to derive meaningful solutions to practical problems.