Solving Equations, Inequalities, and Systems of Equations and Inequalities

In this lesson, we will review solving equations, inequalities, and systems of equations and inequalities. Reviewing this will allow us to also explore modeling in various real-life situation. You have already studied these concepts in previous courses.

Let us first look at solving equations by graphing. Watch the following videos that will help us explore these equations.

In Video 1, the first 03:00 shows how to solve a system of equations by graphing. From 03:00 to 07:55 shows three examples that determine whether a given point is a solution to a system of equations. From 07:55 to the end shows two examples of graphing systems to determine a solution.

In Video 2, the first 03:24 shows how many and what type of solutions a system of equations has, given the graphs of the system of equations: no solution, one solution, and infinitely many solutions. From 03:25 until the end of the video it shows three examples that graph a system and use the system to determine a solution.

Sometimes there is a need to take an equation or formula and solve for a different variable, due to special situations. Watch the following videos that will help you remember the process and some special situation.

Next let's look at rational functions, and how we model real-life situations with these types of functions. Watch the following videos that will help us explore these situations.

In Videos 1 and 2, an example of a modeling with rational functions problem is explored. In this example, the problem explores the number of animals in a pet store.

In Videos 3 and 4, an example of work with combined rates problem is explored. In this example, the work being done is raking lawns.

In Video 5, a second example of work with combined rates problem is explored. In this example, the work being done is by two different hoses.

Next, let's explore the solving of systems of equations by substitution and elimination. Watch the following videos that will help us explore these systems.

In Videos 1, the first 05:08 shows solving a system of equations by the "substitution method." From 05:09 to the end, shows solving a system of equations by the "elimination method." At approximately 10:00 in the video, how to write the solution set is discussed.

In Videos 2, the first 03:18 explores a modeling example of taxi service companies, where solving a system of equations is constructed and used to attain the solution to the problem. From 03:19 until the end, a modeling example of a school drama performance's costs are explored, where solving a system of equations is constructed and used to attain the solution to the problem.

Next, let's explore the solving of systems of equations in three variables. Watch the following videos that will help us explore these systems.

Video 1 shows how to solve a system of equations in three equations by the "substitution method." Video 2 shows how to solve a system of equations in three equations by the "elimination method."

Both videos also show you how to check the solutions by using the TI-84 calculator, especially using matrices.

Next, let's review solving of linear inequalities. Watch the following videos that will help us review this concept.

Video 1 shows four examples on how to solve linear equalities. It also explores how to find the solution using a TI-84 calculator.

In the first 2:36 of video 2, two examples on how to solve linear equalities are shown: a horizontal line example and a vertical line example. From 2:37 to the end of the video, two more examples on how to solve systems of linear equalities are explored. Video 2 also explores how to find the solution set using a TI-84 calculator.

Next, let's explore the solving of systems of inequalities that will help us model real-life situations. Watch the following videos that will help us explore these situations.

In the first 5:03 of video 1, two examples on how to solve systems of linear equalities are explored. It specifically looks at how to graph and solve systems when the words "and" and "or" are used, as well as when you might have "no solution." From 5:04 until the end of the video, an application problem is explored.

Video 2 explores an application problem. It also shows how software, and a TI-84 calculator, can help solve and find the critical intersection points.

Video 3 explores a second application problem. This example, like in video 2, also shows how software, and a TI-84 calculator, can help solve and find the critical intersection points.

One important concept in modeling is to be able to rearrange equations for different variables. Here is a fairly simple example-

Jeff wants to fence in several small rectangular pens for his horses. Because of the lengths of the fence panels that he has and the distance along the property line, he wants the pens to have these widths - 30 feet, 50 feet, and 90 feet. For his purposes, the areas of the pens must be 2400 sq ft, 5000 sq ft, and 11430 sq ft respectively. What will be the lengths of the pens?

The formula for area of a rectangle is equation image indicator LaTeX: A=lw . To find the lengths, Jeff would have to solve 3 equations. Instead, he could rearrange the formula so the LaTeX: l is by itself; then he just has to plug in the other values. The process of solving an equation for a specific variable is the same as solving any equation. Here, Jeff would divide both sides of the equation by w to get equation image indicator $LaTeX: l=\frac{A}{w}$ .

What will be the lengths of the pens?

Solution: 80 feet, 100 feet, and 127 feet

Example 1

When sending a rectangular package through the U.S. Postal Service, the combined length and girth (perimeter of the cross section) cannot exceed 108 inches.

a. What shape is the girth?

Solution: Rectangle or square

b. How do you find the perimeter of that shape?

Solution: Add all the sides or 2l+2w

c. Can you send a box that is equation image indicator $LaTeX: 20\times20\times25$ inches?

Solution: Yes, the girth is 80 plus the length of 25 is 105 inches

d. Does it matter which way you take the girth?

Solution: Yes, if you did the girth the long way you would get 90, plus 20 is 110

Example 2

Mary (from Topic 1 who owns the wedding dress boutique) needs open-topped boxes to store her excess inventory at year's end. Mary purchases large rectangles of thick cardboard with a length of 78 inches and width of 42 inches to make the boxes. Mary is interested in maximizing the volume of the boxes and wants to know what size squares to cut out at each corner of the cardboard (which will allow the corners to be folded up to form the box) in order to do this.

image of rectangle with angles x and length of 78 inches, width of 42 inches

(a) Volume is a three-dimensional measure. What is the third dimension that the value x represents?

Solution: x represents the height of the box

(b) Using the table below, choose five values of x and find the corresponding volumes.

Solution: Answers vary. Realize that the length is 78-2x and the width is 42-2x since you are taking x from each corner. Multiply length, width, and height to get volume.

x	Length	Width	Volume

You tested several different values of x above, and calculated five different volumes. There is a way to guarantee that you use dimensions that will maximize volume, and now we're going to work through that process.

(c) Write an equation for volume in terms of the three dimensions of the box.

Solution: $LaTeX: V=x\left(78-2x\right)\left(42-2x\right) \\ V=x\left(3276-240x+4x^2\right)\\ V=3276x-240x^2+4x^3\\$

(d) Graph the equation from part (c). Remember to adjust your window, thinking about the domain and range. text annotation indicator

Solution: The domain for x is 0 to 21 because if you cut out the 21 inch corners there will be no width left. You should get an idea of the range from the table.

graph of parabola opening down

(e) From your graph, what are the values of the three dimensions that maximize the volume of the box? What is the maximum volume of the box?

Solution: From the graph below, it appears that the maximum occurs at approximately (8.73, 13000), so the maximum volume would be 13000 cubic inches with a height of 8.73 inches, a length of 60.54 inches, and a width of 24.54 inches.

graph of parabola opening down with its apex at (8.73, 1.3E+4)

Example 3

image of 5 foot person looking at a 38 degree angle at a tree with the length of 20 feet and the width of 5 and T

As shown in the diagram, you are standing 20 feet away from a tree, and you measure the angle of elevation to be $LaTeX: 38^\circ$ . How tall is the tree?

Solution: The solution depends on your height, as you measure the angle of elevation from your lien of sight. Assume that you are 5 feet tall. The figure shows us that once we find the value of , we have to add 5 feet to this value to find the total height of the triangle. To find , we should use the tangent value- $LaTeX: \tan\:38^\circ=\frac{T}{20};\:T\approx15.63$ so the height of the tree $LaTeX: \approx20.63$ ft.

Example 4

A roller coaster is modeled by the function $LaTeX: r\left(x\right)=-\left(x-2\right)\left(x-3\right)\left(x-1\right)\left(x-5\right)\left(x-6\right)\left(x-6\right)+25$

a. Graph the function (remember to adjust your window).

Solution:

graph of solution

b. What are the heights, to the nearest foot, of each of the humps?

Solution: 55 ft, 49 ft, 25 ft

c. If the amusement park with this roller coaster covers 6 acres which is 242.81 sq dekameters. If there are 5,432 people in the park at noon, how many people per square dekameter is that, to the nearest tenth? ( a dekameter is 10 meters)

Solution: 22.4 people per dekameter

It's now time for us to explore and practice working Literal Equations.

It's now time for us to explore and practice working Review Modeling of Functions.

Please review the vocabulary, teaching videos, practice problems in these lessons before taking the Functions and Graphs Quiz.

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