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Unit 1: linear equations and inequalities, unit 2: graphs and forms of linear equations, unit 3: functions, unit 4: quadratics: multiplying and factoring, unit 5: quadratic functions and equations, unit 6: complex numbers, unit 7: exponents and radicals, unit 8: rational expressions and equations, unit 9: relating algebra and geometry, unit 10: polynomial arithmetic, unit 11: advanced function types, unit 12: transformations of functions, unit 13: rational exponents and radicals, unit 14: logarithms.
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- 4.2 Modeling with Linear Functions
- Introduction to Prerequisites
- 1.1 Real Numbers: Algebra Essentials
- 1.2 Exponents and Scientific Notation
- 1.3 Radicals and Rational Exponents
- 1.4 Polynomials
- 1.5 Factoring Polynomials
- 1.6 Rational Expressions
- Key Equations
- Key Concepts
- Review Exercises
- Practice Test
- Introduction to Equations and Inequalities
- 2.1 The Rectangular Coordinate Systems and Graphs
- 2.2 Linear Equations in One Variable
- 2.3 Models and Applications
- 2.4 Complex Numbers
- 2.5 Quadratic Equations
- 2.6 Other Types of Equations
- 2.7 Linear Inequalities and Absolute Value Inequalities
- Introduction to Functions
- 3.1 Functions and Function Notation
- 3.2 Domain and Range
- 3.3 Rates of Change and Behavior of Graphs
- 3.4 Composition of Functions
- 3.5 Transformation of Functions
- 3.6 Absolute Value Functions
- 3.7 Inverse Functions
- Introduction to Linear Functions
- 4.1 Linear Functions
- 4.3 Fitting Linear Models to Data
- Introduction to Polynomial and Rational Functions
- 5.1 Quadratic Functions
- 5.2 Power Functions and Polynomial Functions
- 5.3 Graphs of Polynomial Functions
- 5.4 Dividing Polynomials
- 5.5 Zeros of Polynomial Functions
- 5.6 Rational Functions
- 5.7 Inverses and Radical Functions
- 5.8 Modeling Using Variation
- Introduction to Exponential and Logarithmic Functions
- 6.1 Exponential Functions
- 6.2 Graphs of Exponential Functions
- 6.3 Logarithmic Functions
- 6.4 Graphs of Logarithmic Functions
- 6.5 Logarithmic Properties
- 6.6 Exponential and Logarithmic Equations
- 6.7 Exponential and Logarithmic Models
- 6.8 Fitting Exponential Models to Data
- Introduction to Systems of Equations and Inequalities
- 7.1 Systems of Linear Equations: Two Variables
- 7.2 Systems of Linear Equations: Three Variables
- 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
- 7.4 Partial Fractions
- 7.5 Matrices and Matrix Operations
- 7.6 Solving Systems with Gaussian Elimination
- 7.7 Solving Systems with Inverses
- 7.8 Solving Systems with Cramer's Rule
- Introduction to Analytic Geometry
- 8.1 The Ellipse
- 8.2 The Hyperbola
- 8.3 The Parabola
- 8.4 Rotation of Axes
- 8.5 Conic Sections in Polar Coordinates
- Introduction to Sequences, Probability and Counting Theory
- 9.1 Sequences and Their Notations
- 9.2 Arithmetic Sequences
- 9.3 Geometric Sequences
- 9.4 Series and Their Notations
- 9.5 Counting Principles
- 9.6 Binomial Theorem
- 9.7 Probability
In this section, you will:
- Build linear models from verbal descriptions.
- Model a set of data with a linear function.
Elan is a college student who plans to spend a summer in Seattle. Elan has saved $3,500 for their trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent this situation? What would be the x -intercept, and what can Elan learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.
Building Linear Models from Verbal Descriptions
When building linear models to solve problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:
- Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
- Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
- Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
- Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
- When needed, write a formula for the function.
- Solve or evaluate the function using the formula.
- Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
- Clearly convey your result using appropriate units, and answer in full sentences when necessary.
Now let’s take a look at the student in Seattle. In Elan’s situation, there are two changing quantities: time and money. The amount of money they have remaining while on vacation depends on how long they stay. We can use this information to define our variables, including units.
So, the amount of money remaining depends on the number of weeks: M ( t ) M ( t ) .
Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.
The rate of change is constant, so we can start with the linear model M ( t ) = m t + b . M ( t ) = m t + b . Then we can substitute the intercept and slope provided.
To find the t- intercept (horizontal axis intercept), we set the output to zero, and solve for the input.
The t -intercept (horizontal axis intercept) is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Elan will have no money left after 8.75 weeks.
When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before Elan saved $3,500, but the scenario discussed poses the question once they saved $3,500 because this is when the trip and subsequent spending starts. It is also likely that this model is not valid after the t -intercept (horizontal axis intercept), unless Elan uses a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is 0 ≤ t ≤ 8.75. 0 ≤ t ≤ 8.75.
In this example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.
Using a Given Intercept to Build a Model
Some real-world problems provide the vertical axis intercept, which is the constant or initial value. Once the vertical axis intercept is known, the t -intercept (horizontal axis intercept) can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The y -intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.
Now we can set the function equal to 0, and solve for x x to find the x -intercept.
The x -intercept is the number of months it takes her to reach a balance of $0. The x -intercept is 4 months, so it will take Hannah four months to pay off her loan.
Using a Given Input and Output to Build a Model
Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.
Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem.
- Identify the input and output values.
- Convert the data to two coordinate pairs.
- Find the slope.
- Write the linear model.
- Use the model to make a prediction by evaluating the function at a given x -value.
- Use the model to identify an x -value that results in a given y -value.
- Answer the question posed.
Using a Linear Model to Investigate a Town’s Population
A town’s population has been growing linearly. In 2004, the population was 6,200. By 2009, the population had grown to 8,100. Assume this trend continues.
- ⓐ Predict the population in 2013.
- ⓑ Identify the year in which the population will reach 15,000.
The two changing quantities are the population size and time. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the y -intercept would correspond to the year 0, more than 2000 years ago!
To make computation a little nicer, we will define our input as the number of years since 2004.
To predict the population in 2013 ( t = 9 t = 9 ), we would first need an equation for the population. Likewise, to find when the population would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an equation, we need the initial value and the rate of change, or slope.
To determine the rate of change, we will use the change in output per change in input.
The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspond to t = 0 , t = 0 , giving the point ( 0 , 6200 ) . ( 0 , 6200 ) . Notice that through our clever choice of variable definition, we have “given” ourselves the y -intercept of the function. The year 2009 would correspond to t = 5, t = 5, giving the point ( 5 , 8100 ) . ( 5 , 8100 ) .
The two coordinate pairs are ( 0 , 6200 ) ( 0 , 6200 ) and ( 5 , 8100 ) . ( 5 , 8100 ) . Recall that we encountered examples in which we were provided two points earlier in the chapter. We can use these values to calculate the slope.
We already know the y -intercept of the line, so we can immediately write the equation:
To predict the population in 2013, we evaluate our function at t = 9. t = 9.
If the trend continues, our model predicts a population of 9,620 in 2013.
To find when the population will reach 15,000, we can set P ( t ) = 15000 P ( t ) = 15000 and solve for t . t .
Our model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the year 2027.
A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.
ⓐ Write a linear model to represent the cost C C of the company as a function of x , x , the number of doughnuts produced. ⓑ Find and interpret the y -intercept.
A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.
- ⓐ Predict the population in 2014.
- ⓑ Identify the year in which the population will reach 54,000.
Using a Diagram to Build a Model
It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.
Using a Diagram to Model Distance Walked
Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?
In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question:
"How long will it take them to be 2 miles apart"?
In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart. We can see that time will be our input variable, so we’ll define our input and output variables.
Because it is not obvious how to define our output variable, we’ll start by drawing a picture such as Figure 2 .
Initial Value: They both start at the same intersection so when t = 0 , t = 0 , the distance traveled by each person should also be 0. Thus the initial value for each is 0.
Rate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of change. The slope for A A is 4 and the slope for E E is 3.
Using those values, we can write formulas for the distance each person has walked.
For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the “starting point” at the intersection where they both started. Then we can use the variable, A , A , which we introduced above, to represent Anna’s position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable, E , E , to represent Emanuel’s position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure.
We can then define a third variable, D , D , to be the measurement of the distance between Anna and Emanuel. Showing the variables on the diagram is often helpful, as we can see from Figure 3 .
Recall that we need to know how long it takes for D , D , the distance between them, to equal 2 miles. Notice that for any given input t , t , the outputs A ( t ) , E ( t ) , A ( t ) , E ( t ) , and D ( t ) D ( t ) represent distances.
Figure 2 shows us that we can use the Pythagorean Theorem because we have drawn a right angle.
Using the Pythagorean Theorem, we get:
In this scenario we are considering only positive values of t , t , so our distance D ( t ) D ( t ) will always be positive. We can simplify this answer to D ( t ) = 5 t . D ( t ) = 5 t . This means that the distance between Anna and Emanuel is also a linear function. Because D D is a linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the output D ( t ) = 2 D ( t ) = 2 and solve for t . t .
They will fall out of radio contact in 0.4 hour, or 24 minutes.
Should I draw diagrams when given information based on a geometric shape?
Yes. Sketch the figure and label the quantities and unknowns on the sketch.
Using a Diagram to Model Distance Between Cities
There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough?
It might help here to draw a picture of the situation. See Figure 4 . It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates ( 3 0 , 1 0 ) , ( 3 0 , 1 0 ) , and Eastborough at ( 2 0 , 0 ) . ( 2 0 , 0 ) .
Using this point along with the origin, we can find the slope of the line from Westborough to Agritown.
Now we can write an equation to describe the road from Westborough to Agritown.
From this, we can determine the perpendicular road to Eastborough will have slope m = – 3. m = – 3. Because the town of Eastborough is at the point (20, 0), we can find the equation.
We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal,
The roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to the junction.
One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This means real-world applications discussing maps need linear functions to model the distances between reference points.
There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson?
Modeling a Set of Data with Linear Functions
Real-world situations including two or more linear functions may be modeled with a system of linear equations . Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in Figure 5 .
Given a situation that represents a system of linear equations, write the system of equations and identify the solution.
- Identify the input and output of each linear model.
- Identify the slope and y -intercept of each linear model.
- Find the solution by setting the two linear functions equal to another and solving for x , x , or find the point of intersection on a graph.
Building a System of Linear Models to Choose a Truck Rental Company
Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile 4 . When will Keep on Trucking, Inc. be the better choice for Jamal?
The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions in Table 1 .
A linear function is of the form f ( x ) = m x + b . f ( x ) = m x + b . Using the rates of change and initial charges, we can write the equations
Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when K ( d ) < M ( d ) . K ( d ) < M ( d ) . The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the K ( d ) K ( d ) function is smaller.
These graphs are sketched in Figure 6 , with K ( d ) K ( d ) in blue.
To find the intersection, we set the equations equal and solve:
This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that K ( d ) K ( d ) is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is d > 100 d > 100 .
Access this online resource for additional instruction and practice with linear function models.
- Interpreting a Linear Function
4.2 Section Exercises
Explain how to find the input variable in a word problem that uses a linear function.
Explain how to find the output variable in a word problem that uses a linear function.
Explain how to interpret the initial value in a word problem that uses a linear function.
Explain how to determine the slope in a word problem that uses a linear function.
Find the area of a parallelogram bounded by the y -axis, the line x = 3 , x = 3 , the line f ( x ) = 1 + 2 x , f ( x ) = 1 + 2 x , and the line parallel to f ( x ) f ( x ) passing through ( 2 , 7 ) . ( 2 , 7 ) .
Find the area of a triangle bounded by the x -axis, the line f ( x ) = 12 – 1 3 x , f ( x ) = 12 – 1 3 x , and the line perpendicular to f ( x ) f ( x ) that passes through the origin.
Find the area of a triangle bounded by the y -axis, the line f ( x ) = 9 – 6 7 x , f ( x ) = 9 – 6 7 x , and the line perpendicular to f ( x ) f ( x ) that passes through the origin.
Find the area of a parallelogram bounded by the x -axis, the line g ( x ) = 2 , g ( x ) = 2 , the line f ( x ) = 3 x , f ( x ) = 3 x , and the line parallel to f ( x ) f ( x ) passing through ( 6 , 1 ) . ( 6 , 1 ) .
For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped to 4,700. Assume this trend continues.
Predict the population in 2016.
Identify the year in which the population will reach 0.
For the following exercises, consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.
Identify the year in which the population will reach 75,000.
For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.
Find the linear function that models the town’s population P P as a function of the year, t , t , where t t is the number of years since the model began.
Find a reasonable domain and range for the function P . P .
If the function P P is graphed, find and interpret the x - and y -intercepts.
If the function P P is graphed, find and interpret the slope of the function.
When will the population reach 100,000?
What is the population in the year 12 years from the onset of the model?
For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.
Find the linear function that models the baby’s weight W W as a function of the age of the baby, in months, t . t .
Find a reasonable domain and range for the function W . W .
If the function W W is graphed, find and interpret the x - and y -intercepts.
If the function W is graphed, find and interpret the slope of the function.
When did the baby weight 10.4 pounds?
What is the output when the input is 6.2?
For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted.
Find the linear function that models the number of people inflicted with the common cold C C as a function of the year, t . t .
Find a reasonable domain and range for the function C . C .
If the function C C is graphed, find and interpret the x - and y -intercepts.
If the function C C is graphed, find and interpret the slope of the function.
When will the output reach 0?
In what year will the number of people be 9,700?
For the following exercises, use the graph in Figure 7 , which shows the profit, y , y , in thousands of dollars, of a company in a given year, t , t , where t t represents the number of years since 1980.
Find the linear function y , y , where y y depends on t , t , the number of years since 1980.
Find and interpret the y -intercept.
Find and interpret the x -intercept.
Find and interpret the slope.
For the following exercises, use the graph in Figure 8 , which shows the profit, y , y , in thousands of dollars, of a company in a given year, t , t , where t t represents the number of years since 1980.
For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in Table 2 . Assume that the house values are changing linearly.
In which state have home values increased at a higher rate?
If these trends were to continue, what would be the median home value in Mississippi in 2010?
If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd.)
For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in Table 3 . Assume that the house values are changing linearly.
If these trends were to continue, what would be the median home value in Indiana in 2010?
In 2004, a school population was 1001. By 2008 the population had grown to 1697. Assume the population is changing linearly.
- ⓐ How much did the population grow between the year 2004 and 2008?
- ⓑ How long did it take the population to grow from 1001 students to 1697 students?
- ⓒ What is the average population growth per year?
- ⓓ What was the population in the year 2000?
- ⓔ Find an equation for the population, P , P , of the school t years after 2000.
- ⓕ Using your equation, predict the population of the school in 2011.
In 2003, a town’s population was 1431. By 2007 the population had grown to 2134. Assume the population is changing linearly.
- ⓐ How much did the population grow between the year 2003 and 2007?
- ⓑ How long did it take the population to grow from 1431 people to 2134 people?
- ⓔ Find an equation for the population, P , P , of the town t t years after 2000.
- ⓕ Using your equation, predict the population of the town in 2014.
A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used for voice or video calling. If a customer uses 410 minutes, the monthly cost will be $71.50. If the customer uses 720 minutes, the monthly cost will be $118.
- ⓐ Find a linear equation for the monthly cost of the cell plan as a function of x , the number of monthly minutes used.
- ⓑ Interpret the slope and y -intercept of the equation.
- ⓒ Use your equation to find the total monthly cost if 687 minutes are used.
A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be $17.80.
- ⓐ Find a linear equation for the monthly cost of the data plan as a function of x , x , the number of MB used.
- ⓒ Use your equation to find the total monthly cost if 250 MB are used.
In 1991, the moose population in a park was measured to be 4,360. By 1999, the population was measured again to be 5,880. Assume the population continues to change linearly.
- ⓐ Find a formula for the moose population, P since 1990.
- ⓑ What does your model predict the moose population to be in 2003?
In 2003, the owl population in a park was measured to be 340. By 2007, the population was measured again to be 285. The population changes linearly. Let the input be years since 2003.
- ⓐ Find a formula for the owl population, P . P . Let the input be years since 2003.
- ⓑ What does your model predict the owl population to be in 2012?
The Federal Helium Reserve held about 16 billion cubic feet of helium in 2010 and is being depleted by about 2.1 billion cubic feet each year.
- ⓐ Give a linear equation for the remaining federal helium reserves, R , R , in terms of t , t , the number of years since 2010.
- ⓑ In 2015, what will the helium reserves be?
- ⓒ If the rate of depletion doesn’t change, in what year will the Federal Helium Reserve be depleted?
Suppose the world’s oil reserves in 2014 are 1,820 billion barrels. If, on average, the total reserves are decreasing by 25 billion barrels of oil each year:
- ⓐ Give a linear equation for the remaining oil reserves, R , R , in terms of t , t , the number of years since now.
- ⓑ Seven years from now, what will the oil reserves be?
- ⓒ If the rate at which the reserves are decreasing is constant, when will the world’s oil reserves be depleted?
You are choosing between two different prepaid cell phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $19.95 plus 11 cents per minute. How many minutes would you have to use in a month in order for the second plan to be preferable?
You are choosing between two different window washing companies. The first charges $5 per window. The second charges a base fee of $40 plus $3 per window. How many windows would you need to have for the second company to be preferable?
When hired at a new job selling jewelry, you are given two pay options:
Option A: Base salary of $17,000 a year with a commission of 12% of your sales
Option B: Base salary of $20,000 a year with a commission of 5% of your sales
How much jewelry would you need to sell for option A to produce a larger income?
When hired at a new job selling electronics, you are given two pay options:
Option A: Base salary of $14,000 a year with a commission of 10% of your sales
Option B: Base salary of $19,000 a year with a commission of 4% of your sales
How much electronics would you need to sell for option A to produce a larger income?
Option A: Base salary of $20,000 a year with a commission of 12% of your sales
Option B: Base salary of $26,000 a year with a commission of 3% of your sales
Option A: Base salary of $10,000 a year with a commission of 9% of your sales
Option B: Base salary of $20,000 a year with a commission of 4% of your sales
- 4 Rates retrieved Aug 2, 2010 from http://www.budgettruck.com and http://www.uhaul.com/
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Access for free at https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
- Authors: Jay Abramson
- Publisher/website: OpenStax
- Book title: College Algebra 2e
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Looking for FREE printable CLEP College Algebra worksheets to help you review the math concepts? Here's a collection of PDF worksheets for all CLEP College Algebra topics.
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CLEP College Algebra Prep 2020 A Comprehensive Review and Step-By-Step Guide to Preparing for the CLEP College Algebra Test
Fundamentals and building blocks.
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Equations and Inequalities
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- Multiplying Binomials
- Factoring Trinomials
- Operations with Polynomials
- Solving a Quadratic Equation
- Quadratic Formula and the Discriminant
- Graphing Quadratic Functions
- Solving Quadratic Inequalities
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Sequences and Series
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Best 30 Algebra Books That Experts Highly Recommend
Learning algebra can be difficult and confusing, but it is an important skill to learn. Indeed, it is a great life skill that can help you in the future. However, learning algebra can be a lot easier if you have a good algebra book. Moreover, Algebra books often have algebra examples, step-by-step instructions, and many different problems for you to solve.
Well, choosing the right book for Algebra can be a difficult decision. Although, there are many different options available. In particular, we have put together a guide of the best books on algebra by skill level and test to guide you in this decision.
On the other hand, if you are looking for an algebra homework helper, don’t worry you can get the best algebra homework help from our experts. So, what are you waiting for get the best help now!
Importance Of Algebra In Today’s World
Table of Contents
Algebra is a very important part of mathematics and is used in the sciences and the world of practical applications of linear algebra . However, it’s easy to see the importance of algebra in school, and learning the basics of math early in life will help a student throughout their education.
Besides, the implications of algebra in the outside world are vast. Further, the use of algebra in many fields such as accounting, engineering, science, and medicine.
Do You Actually Need to Learn About Algebra Concepts?
This is a basic question that always raises in the average student’s mind. But yes, it is true that algebra makes your life’s calculations quite simpler. How? Let’s find out!
So, how does algebra come into play in everyday life? Assume you are requested to purchase eggs to decorate for Easter. You have 5 bucks and rush to the supermarket, where you discover that a carton of a dozen eggs costs $1.30. You can calculate how many eggs you can buy through a variety of methods, but because you’ve studied mathematics, your mind can answer this issue quickly.
To arrive at an answer, mentally divide the entire dollar amount by the cost of eggs. Here, algebra tells you that you can’t go beyond $5 utilizing the notion of inequalities.
Because you know you can only buy 3 cartons of eggs, you’ll walk up to the cashier with the correct number of cartons of eggs and won’t have to worry about overpaying. This is a simple example.
Another application of algebra in real life is with functions. Functions represent information linkages, similar to cause and effect. Although you may not believe you utilize functions at all, I can assure you that you do them virtually every day.
For example, you may be aware that when you microwave a glass of water, the longer you heat it, the hotter the water becomes. You know that the temperature of the water is a function of the time it is microwaved because of your understanding of functions.
Numerous real-life situations may be represented using functions and other algebraic ideas. Learning algebraic concepts helped to clarify your thinking by improving your understanding of these real-world situations and your ability to draw linkages between the facts.
How can Algebra books help?
There are many ways that algebra books can help people who need to work on their algebra. For example, these books are a great way to introduce yourself to new topics that you may not have learned in class. There are also lots of practice problems and exercises in these books, which you find helpful in reaching my goals with algebra.
Furthermore, Algebra textbooks are a great way to improve how well you understand the subject and can make learning easier.
Modes Of Availability of Algebra Books
There are two ways to availability of the best algebra books. Firstly, you can access these books through an online platform; Secondly, you can access books through an offline one.
Here are the best textbooks that are available in algebra books online mode.
1. Regents Algebra I Power Pack Revised Edition
Writer: Gary M. Rubinstein M.S
Rating: ☆☆☆☆☆ (5 out of 5)
Online source: Kindle Unlimited, Scribd
Regents Algebra I Power Pack Revised Edition is an excellent book for a student who is interested in learning algebra. This book will cover finding x as well as polynomial functions and equations, linear equations, and quadratic equations.
Further, this book is extremely well organized, covering topics in perfect order. Besides, this book is full of exercises to help you understand and review the material.
2. Algebra Essentials Practice Workbook with Answers
Writer: Chris McMullen
Indeed, Algebra Essentials Practice Workbook with Answers is one of the best algebra books online. However, it is designed to help students with understanding the basic concepts of algebra and further their knowledge of the subject.
Moreover, it can help students if they are learning on their own or if they are taking a course in algebra.
3. Pre-Algebra: Order of Operations
Writer: Humble Math
Pre-Algebra: Order of Operations is also a helpful book to read if you’re having trouble solving problems with the order of operations. Moreover, this book is a vital concept to learn in math. It teaches the order in which mathematical operations should be solved.
In addition, the author is very easy to read and presents everything in a very logical order.
Here are some of the top best author’s books on Algebra which are not available in online mode. You can also read these books offline.
1. Richard W. Fisher
Book: No-Nonsense Algebra
Richard W. Fisher has written many books on algebra, but No-Nonsense Algebra is the best book. In addition, these books are helpful for students who are preparing for their high school algebra exams.
Indeed, Rick Fisher was a math instructor for the Oak Grove School District in San Jose, California, for over 31 years. Since graduating from San Jose State University in 1971 with a B.A. in mathematics, Rick has devoted his time to both teaching and developing unique award-winning math materials.
Buy & read this book:
This book is available in many libraries as well as on Amazon. You can also read it sitting at home by ordering it from Amazon.
2. Gilbert Strang
Book: Introduction to Linear Algebra
Indeed, Gilbert Strang is a professor of mathematics at the Massachusetts Institute of Technology, where his research focuses on analysis, linear algebra and PDEs.
Indeed, Gilbert Strang is an excellent writer for algebra. He also wrote about linear algebra, multivariable calculus, and calculus solutions.
The Linear Algebra book by L. LeBlanc is available at most bookstores across the world. Furthermore, you can also buy this book online at sites like amazon.com, which has a rather good price.
3. Lynette Long
Book: Painless Algebra
Lynette Long is a licensed psychologist and former college professor. Indeed, she has published more than twenty-five books and dozens of articles and has written two award-winning plays. A math education expert, she is particularly interested in the math achievement of young girls.
The Painless Algebra book is available at most bookstores as well as libraries. Moreover, you can also buy this book at amazon.com.
Best Algebra Books for Beginners and High School
Here are some of the best algebra books for beginners and high school:
Check Top Algebra Textbooks For Beginners
1. introduction to abstract algebra.
Writer: Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger
Introduction to Abstract Algebra is the best book for algebra.
This is a tremendously good book, presented in larger type, which makes it extremely pleasant to read. Additionally, this book helps beginners fully understand groups, rings, semigroups, and monoids by rigorously building concepts from the first principles.
About the Author:
In fact, Benjamin Fine is a professor of mathematics at Fairfield University.
Indeed, Anthony M. Gaglione is a professor of mathematics at the United States Naval Academy.
Gerhard Rosenberger is a professor of mathematics at the University of Hamburg.
2. Abstract Algebra: An Interactive Approach
Author: William Paulsen
Abstract Algebra: An Interactive Approach is a book for algebra.
However, the textbook gives an introduction to algebra . Furthermore, this book teaches how students can better grasp difficult algebraic concepts through the use of computer programs.
In addition, this book encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area.
Indeed, it is also an excellent book of algebra for beginners if you are just starting out.
In fact, William Paulsen, PhD, professor of mathematics, Arkansas State University, USA
3. An Introduction to Essential Algebraic Structures
Author: Martyn R. Dixon, PhD, Leonid A. Kurdachenko, Igor Ya. Subbotin
Rating: ☆☆☆☆ (4 out of 5)
An Introduction to Essential Algebraic Structures teaches an integrated approach to basic concepts of modern algebra. This book also highlights topics that play a major role in various branches of mathematics.
Further, this book begins with a discussion of set theory’s fundamental concepts before moving on to abstract algebra’s core notions and branches. Overall, this textbook is the best book for algebra.
Firstly, Martyn R. Dixon, PhD, is a Professor in the Department of Mathematics at the University of Alabama.
Secondly, Leonid A. Kurdachenko, PhD, is a Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine.
Thirdly, Igor Ya. Subbotin, PhD, is a Professor in the Department of Natural Sciences and Mathematics at the National University in Los Angeles, California.
Furthermore, there are some other books which are very useful for beginners.
4. Abstract Algebra
Author: Paul B. Garrett
5. Pure Mathematics for Beginners
Author: Steve Warner
6. Introduction to Relation Algebras
7. A History of Abstract Algebra
Author: Jeremy Gray
Check Top Algebra Books For High School
Here are the top books about algebra for high school.
1. High School Algebra I Unlocked
Author: Princeton Review
High School Algebra I Unlocked focuses on giving you a wide range of key techniques to help you tackle subjects like Algebra.
However, you’ll discover the link between abstract concepts and their real-world applications and build confidence as your skills improve with this book.
Additionally, you’ll get plenty of practice, from fully guided examples to independent end-of-chapter drills and test-like samples.
The experts at The Princeton Review have been helping students, parents, and educators achieve the best results at every stage of the education process since 1981.
2. Algebra I Workbook For Dummies
Author: Mary Jane Sterling
This book is the solution to the Algebra brain block. In addition, with hundreds of practice and example problems mapped to the typical high school Algebra class, this book teaches how to crack the code in no time!
However, this book is the best way to learn algebra is to work on the problems and let the numbers fly.
About the author:
Indeed, Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of Algebra I For Dummies and Algebra II For Dummies.
3. McGraw-Hill Education Algebra I Review and Workbook
Author: Sandra Luna McCune
McGraw-Hill Education Algebra I Review and Workbook is one of the best books on algebra.
However, this book is very concise. Besides, this book is great for high-school students.
Furthermore, this book introduces concepts well and is fairly comprehensive. With this book, you will also learn how to apply Algebra I to practical situations.
About the author:
Indeed, Sandra Luna McCune, PhD, is a former Regents Professor who taught as a mathematics specialist at Stephen F. Austin State University.
Moreover, there are some other algebra books for high school that are very useful.
4. Practice Makes Perfect Algebra I Review and Workbook
Author: Carolyn Wheater
5. 101 Involved Algebra Problems with Answers
Author: Chris McMullen
6. Must Know High School Algebra
Author: Chris Monahan
7. Algebra II: Essential Practice for Advanced Math Topics
Author: Inc. Carson-Dellosa Publishing
Top 10 college algebra books
1. College Algebra
Author: James Stewart, Lothar Redlin, Saleem Watson
2. Essentials of College Algebra
Author: Margaret Lial, John Hornsby, David Schneider
3. College Algebra with Intermediate Algebra: A Blended Course
Author: Judith Beecher, Judith Penna, Barbara Johnson
4. College Algebra DeMYSTiFieD
Author: Rhonda Huettenmueller
5. Algebra for College Students
Author: Jerome E. Kaufmann, Karen L. Schwitters
6. College Algebra with Modeling & Visualization
Author: Gary Rockswold
7. College Algebra: Real Mathematics, Real People
Author: Ron Larson
8. College Algebra, Books a la Carte Edition
Author: J.S. Ratti, Marcus McWaters, Leslaw Skrzypek
9. Algebra and Trigonometry with Analytic Geometry
Author: Earl W. Swokowski, Jeffery A. Cole
10. Functions and Change: A Modeling Approach to College Algebra
Author: Bruce Crauder, Benny Evans, Alan Noell
Thus, these are some college algebra books.
We hope you enjoyed our blog about algebra books. Moreover, we know that you can make the most of your algebraic lessons and complete all your homework with ease with this knowledge.
These are very good algebra books that provide not only clear examples but also easy to understand examples. Furthermore, algebra is one of the foundations for math, and it will help students to be prepared for more advanced math courses. Although, a well-formed best algebra textbook will offer you immeasurable support to enhance your expertise in all the topics of algebra.
Frequently Asked Questions
Q1. what are the 4 types of algebra.
There are not four distinct algebra types, but different branches of algebra focus on different types of mathematical structures and techniques. Here are some examples:
1. Elementary algebra: This branch of algebra deals with manipulating and solving algebraic expressions, equations, and inequalities.
2. Abstract algebra: This branch of algebra focuses on the study of algebraic structures such as groups, rings, and fields.
3. Linear algebra: This branch of algebra studies vectors, matrices, and linear transformations.
4. Boolean algebra: This branch of algebra studies Boolean functions, which take binary inputs (0 or 1) and produce binary outputs.
Q2.Why is it important to study algebra?
The ability to absorb complicated, evolving as well as abstract concepts stimulate the brain, allowing students to discover new ways of thinking. Algebra also help students in organizing their thoughts, making it easier for them to come up with appropriate responses in complex or dynamic situations.
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