Linear Equations and Inequalities: Background You’ll Need 3

Giselle Miller

Linear Equations and Inequalities: Background You’ll Need 3

Extract relevant information from word problems and interpret mathematical notation in real-world contexts

In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you will need (and won’t need) to answer the question.

Strategies for Reading and Understanding Math Problems

Reading and understanding math problems is an essential skill for successful problem-solving. Let’s explore effective strategies to help you navigate and comprehend math problems with ease. By applying these strategies, you will develop the ability to extract key information, identify problem objectives, and confidently solve mathematical problems.

1. Read Carefully: Begin by reading the problem statement attentively. Understand the context, identify the problem’s objective, and note any important details or constraints. Pay close attention to numerical values, units of measurement, and keywords that indicate mathematical operations.
2. Identify the Question: Determine what the problem is asking you to find or solve. Look for phrases like “Find,” “Calculate,” or “Determine.” Understanding the question will guide your approach and help you focus on the relevant information.
3. Highlight Key Information: Identify and highlight the essential details and quantities provided in the problem. This includes numerical values, units, and any other relevant data. Use labels or variables to represent unknowns or quantities that need to be calculated. Highlighting the key information allows you to focus on what is important information and ignore unimportant distracting information.
4. Visualize the Problem: Create mental or written representations of the problem to aid your understanding. Draw diagrams, charts, or graphs if appropriate. Visualizing the problem can help you see the relationships between quantities and identify potential solution paths.
5. Break It Down: Break the problem into smaller parts or steps. Analyze each part individually to understand its purpose and how it contributes to the overall solution. Identify the mathematical operations or concepts required for each step.
6. Use Problem-Solving Strategies: Familiarize yourself with problem-solving strategies, such as working backward, making a table or chart, using logical reasoning, or applying relevant formulas or equations. Choose the strategy that best suits the problem at hand. We will explore problem-solving strategies more later in this section.
7. Solve Step-by-Step: Once you have a plan in mind, solve the problem step-by-step. Show your work and perform the necessary calculations, ensuring accuracy and attention to detail.
8. Check Your Answer: After finding a solution, verify if it makes sense in the given context. Re-read the problem, check your calculations, and assess whether the answer aligns with the question’s requirements. If possible, use estimation or alternative methods to confirm the reasonableness of your answer.
9. Reflect on the Solution: Take a moment to reflect on the problem-solving process. Evaluate the effectiveness of your strategy and identify any insights or learnings that can be applied to similar problems in the future.

By employing these strategies, you can improve your ability to read and understand math problems, identify relevant information, and approach problem-solving with clarity and confidence. Regular practice with a variety of problems will enhance your skills and deepen your mathematical understanding.

Classifying the Types of Problems

In mathematics, different types of problems require distinct approaches and techniques to arrive at a solution. Are we trying to solve for an unknown variable? Maybe we need to simplify a complex expression. Or perhaps, we’re asked to calculate a certain quantity. Sometimes, we might even need to graph a function or navigate a multi-step problem with multiple operations and parts to consider. Let’s look at various problem classifications to help you identify the nature of a problem and apply the appropriate problem-solving strategies.

Solve: In these problems, we’re typically trying to find the value of an unknown. For example, “solve for [latex]x[/latex] in the equation [latex]2x + 3 = 7[/latex]“. To solve it, we’ll need to isolate [latex]x[/latex] on one side of the equation.
Simplify: In a simplification problem, we might be given a complicated expression, like [latex](3x^2)^2[/latex]. Here, our job is to simplify it to a more manageable form, in this case, [latex]9x^4[/latex].
Calculate: Calculation problems ask us to compute a specific numerical value. “Calculate the area of a circle with radius [latex]3[/latex]” is an example. Here, we need to use the formula for the area of a circle ([latex]πr^2[/latex]) to get our answer.
Graph: Graphing problems usually involve plotting a function or equation on a coordinate plane. For example, “Graph the function [latex]y = 2x - 1[/latex]“. We would find several values of [latex]y[/latex] for different [latex]x[/latex]-values and plot those points on the graph.
Multi-step: These problems are a mix of the above types and involve multiple steps. They often require careful planning and understanding of the order of operations. For instance, “Solve for [latex]x[/latex] in the equation [latex]2x + 3 = 7[/latex], then calculate the value of [latex]y[/latex] in the equation [latex]y = 3x - 2[/latex]“.

It’s important to remember that these types aren’t mutually exclusive. A problem can have elements of more than one type. For example, a multi-step problem may require us to solve an equation, simplify an expression, and then calculate a final value. Remember, the key to conquering any math problem is understanding what it’s asking you to do. Once you’ve got that down, you’re halfway there!

Strategy Makes The Difference

In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes. This approach does not work well with real-life problems. Instead, problem-solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backward, asking “what information and procedures will I need to find this?”

How To: Problem-Solving Process

Identify the question you’re trying to answer.
Work backward, identifying the information you will need and the relationships you will use to answer that question.
Continue working backward, creating a solution pathway.
If you are missing any necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
Solve the problem, following your solution pathway.

Now that we have a process for problem-solving, let’s talk about the different approaches we can take to solve a problem.

Break It Down: Complex problems can often be daunting. But here’s a secret – they’re just a bunch of simple problems grouped together. Our first approach should always be to break down complex problems into smaller, more manageable parts.
- For example, if you are asked “”A farmer has chickens and cows in his farm. He counts [latex]50[/latex] heads and [latex]140[/latex] legs. How many chickens and cows does he have?” you may think this sounds super complex but if we break it down we see it isn’t so bad
  - All animals have [latex]1[/latex] head. So, the [latex]50[/latex] heads mean we have [latex]50[/latex] animals.
  - Chickens have [latex]2[/latex] legs, cows have [latex]4[/latex]. So if all [latex]50[/latex] animals were chickens, we would have [latex]100[/latex] legs.
  - But we have [latex]140[/latex] legs, which is [latex]40[/latex] more than [latex]100[/latex]. Since each cow has [latex]2[/latex] extra legs compared to a chicken, the [latex]40[/latex] extra legs mean we have [latex]20[/latex] cows ( [latex]40 ÷ 2 = 20[/latex]).
  - Finally, since we have [latex]50[/latex] animals in total, the remaining [latex]30[/latex] must be chickens.
Trial and Error: Sometimes, problems don’t have a clear path to the solution. In these cases, good old trial and error can come to our rescue.
- For example, if you are asked “What is the value of [latex]x[/latex] in the equation [latex]2^x = 32?[/latex]” it may be tempting to jump right into logarithms, but if you try a few values for [latex]x[/latex] first it may help to find the answer.
  - If [latex]x = 4[/latex], then [latex]2^x = 2^4 = 16[/latex]. Too small.
  - If [latex]x = 5[/latex], then [latex]2^x = 2^5 = 32[/latex]. Bingo!
Pattern Recognition: In mathematics, patterns are everywhere! Recognizing these patterns can make problem-solving a breeze.
- For example, if you are asked “What is the [latex]5[/latex]th term in the sequence: [latex]2, 4, 8, 16,[/latex]…?” looking for a pattern will help solve the problem.
  - Here, we can see that each term is twice the previous one. So, the [latex]5[/latex]th term is [latex]16*2 = 32[/latex].
Logical Reasoning: using logical reasoning can be a powerful problem-solving strategy. This involves creating a logical sequence of steps to solve the problem.
- For example, if you are asked “If all squares are rectangles, and all rectangles have four sides, do all squares have four sides?” you can use logic to find a solution
  - We can logically reason that since every square is a rectangle, and every rectangle has four sides, it must be that every square has four sides too.

Let’s try using our problem-solving process and the approaches we just learned to solve a few examples. In the first example, we will need to think about time scales, we are asked to find how many times a heart beats in a year, but usually we measure heart rate in beats per minute.

How many times does your heart beat in a year?

The technique that helped us solve the last problem was to get the number of heartbeats in a minute translated into the number of heartbeats in a year. Converting units from one to another, like minutes to years is a common tool for solving problems.

In the next example, we show how to infer the thickness of something too small to measure with every-day tools. Before precision instruments were widely available, scientists and engineers had to get creative with ways to measure either very small or very large things. Imagine how early astronomers inferred the distance to stars, or the circumference of the earth.

How thick is a single sheet of paper? How much does it weigh?

We can infer a measurement by using scaling. If [latex]500[/latex] sheets of paper is two inches thick, then we could use proportional reasoning to infer the thickness of one sheet of paper.

In the next example, we use proportional reasoning to determine how many calories are in a mini muffin when you are given the amount of calories for a regular sized muffin.

A recipe for zucchini muffins states that it yields [latex]12[/latex] muffins, with [latex]250[/latex] calories per muffin. You instead decide to make mini-muffins, and the recipe yields [latex]20[/latex] muffins. If you eat [latex]4[/latex], how many calories will you consume?

We have found that ratios are very helpful when we know some information but it is not in the right units, or parts to answer our question we have to make comparisons. Making comparisons mathematically often involves using ratios and proportions. In the next examples we will will use proportions and rates to make a comparison.

You need to replace the boards on your deck. About how much will the materials cost?

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.From Hyundai’s website, the 2013 Sonata will get [latex]24[/latex] miles per gallon (mpg) in the city, and [latex]35[/latex] mpg on the highway. The hybrid will get [latex]35[/latex] mpg in the city, and [latex]40[/latex] mpg on the highway. An average driver drives about [latex]12,000[/latex] miles a year.Suppose that you expect to drive about [latex]75\%[/latex] of that in the city, so [latex]9,000[/latex] city miles a year, and [latex]3,000[/latex] highway miles a year. We can then find the number of gallons each car would require for the year.

Sonata: [latex]\displaystyle{9000}\text{ city miles}\cdot\frac{1\text{ gallon}}{24\text{ city miles}}+3000\text{ highway miles}\cdot\frac{1\text{ gallon}}{35\text{ highway miles}}=460.7\text{ gallons}[/latex]

Hybrid: [latex]\displaystyle{9000}\text{ city miles}\cdot\frac{1\text{ gallon}}{35\text{ city miles}}+3000\text{ highway miles}\cdot\frac{1\text{ gallon}}{40\text{ highway miles}}=332.1\text{ gallons}[/latex]

If gas in your area averages about [latex]$3.50[/latex] per gallon, we can use that to find the running cost:

Sonata: [latex]\displaystyle{460.7}\text{ gallons}\cdot\frac{\$3.50}{\text{gallon}}=\$1612.45[/latex]

Hybrid: [latex]\displaystyle{332.1}\text{ gallons}\cdot\frac{\$3.50}{\text{gallon}}=\$1162.35[/latex]

The hybrid will save [latex]$450.10[/latex] a year. The gas costs for the hybrid are about [latex]\displaystyle\frac{\$450.10}{\$1612.45} = 0.279 = 27.9\%[/latex] lower than the costs for the standard Sonata.While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about [latex]$25,850[/latex], compared to the base model for the regular Sonata, at [latex]$20,895[/latex].To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs [latex]$4,965[/latex] more. With gas savings of [latex]$451.10[/latex] a year, it will take about [latex]11[/latex] years for the gas savings to make up for the higher initial costs.We can conclude that if you expect to own the car [latex]11[/latex] years, the hybrid is indeed worth it. If you plan to own the car for less than [latex]11[/latex] years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.To see this example worked out in a video, watch the following:

You can view the transcript for “Guiding decision using math: Sonata vs Hybrid” here (opens in new window).

Try using the problem-solving process and the approaches you learned to solve some questions on your own.