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Linear equations and inequalities

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You are the financial manager of a furniture company. It is your job to create supply and demand graphs that show the break-even point for the company. The functions that describe how the company can break even will be represented with linear equations. Linear inequalities will be used to graph conditions where the company experiences losses and also where the company would be making a profit.

Before you get started with the actual financial numbers, you will want to practice by creating and comparing various kinds of graphs. To start, you will be comparing the graphs of linear equations with those of linear inequalities.

Consider first a linear equation of the form y = mx + b.
1. Choose values of m and b, then plot the initial equation.
2. Change m to make a different plot.
3. Change b (but not m) in the original equation and plot the result.
4. Finally, use the new values of m and b to create a plot. Place all four on the same graph.

Now choose a linear inequality of the form y < mx + b or y > mx + b.
1. Choose values of m and b, then plot the initial equation.
2. Change m to make a different plot.
3. Change b (but not m) in the original equation, and plot the result.
4. Use the new values of m and b to create a plot.
5. Place all four on the same second graph.

Write a summary that compares the plots on the first graph to one another, then the plots on the second graph to one another, and finally the plots on the first graph to the plots on the second graph. Discuss the effects of changing m, b, and the equality or inequality sign.

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This solution contains a detailed explanation of distinguishing linear equations from inequalities.

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Details: You are the financial manager of a furniture company. It is your job to create supply and demand graphs that show the break-even point for the company. The functions that describe how the company can break even will be represented with linear equations. Linear inequalities will be used to graph conditions where the company experiences losses and also where the company would be making a profit.

Before you get started with the actual financial numbers, you will want to practice by creating and comparing various kinds of graphs. To start, you will be comparing the graphs of linear equations with those of linear inequalities.

Consider first a linear equation of the form y = mx + b.
1. Choose values of m and b, then plot ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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