Solving Equations and Inequalities Graphically

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245

Solving Equations and

Inequalities Graphically

4.4

4.4 OBJECTIVES

1. Solve linear equations graphically 2. Solve linear inequalities graphically

In Chapter 2, we solved linear equations and inequalities. In this section, we will graphi-cally demonstrate solutions for similar statements. In using this section, note that each graphical demonstration is accompanied by the algebraic solution, which appears in the margin. The techniques of this section are not designed as an alternative to the algebra. They are rather an introduction to the idea of “viewing a solution.” This is a skill that will be very useful as you continue to study mathematics.

In our first example, we will solve a simple linear equation. The graphical method may seem cumbersome, but once you master it, you will find it quite helpful, particularly if you are a visual learner.

A Graphical Approach to Solving a Linear Equation

Graphically solve the following equation. 2x60

Step 1 Let each side of the equation represent a function of x.

f(x)2x6

g(x)0

Step 2 Graph the two functions on the same set of axes.

Step 3 Find the intersection of the two graphs. To do this, examine the two graphs

closely to see where they intersect. Identify the coordinates of that point. This intersection determines the solution to the original equation.

g y f Example 1 NOTEAlgebraically 2x60 2x6 x3

NOTEWe ask the question, “when is the graph of fequal to the graph of g?” Specifically, for what values of xdoes this occur?

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246 CHAPTER4 GRAPHS OFLINEAREQUATIONS ANDFUNCTIONS

The two lines intersect on the xaxis at the point (3, 0). We are looking for the xvalue at the point of intersection, which is 3.

g

y f

(3, 0)

C H E C K Y O U R S E L F 1

Graphically solve the following equation.

3x60

The graph of the equation is often used to check the algebraic solution. This concept is illustrated in Example 2.

Example 2

Solving Linear Equations Algebraically and Graphically

Solve the linear equation algebraically, then graphically display the solution. 2(x3) 3x4

To graphically display the solution, let

f(x)2(x3)

g(x) 3x4

Graphing both lines, we get

g f x y NOTEAlgebraically 2x6 3x4 5x6 4 5x 10 x 2

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SOLVINGEQUATIONS ANDINEQUALITIESGRAPHICALLY SECTION4.4 247

The point of intersection appears to be (2, 2), which confirms that 2 is a reasonable solution to the equation

2(x3) 3x4

First solve the linear equation algebraically, then graphically display the solution. 3x42x1

The following algorithm summarizes our work in graphically solving an equation.

Step 1 Let each side of the equation represent a function of x. Step 2 Graph the two functions on the same set of axes.

Step 3 Find the intersection of the two graphs. The xvalue at this intersection represents the solution to the original equation.

Step by Step: Graphically Solving an Equation

We will now use the graphs of linear functions to determine the solutions of a linear in-equality.

Linear inequalities in one variable, x, are obtained from linear equations by replacing the symbol for equality () with one of the inequality symbols (,,, ).

The general form for a linear inequality in one variable is

xa

in which the symbol can be replaced with ,, or . Examples of linear inequalities in one variable include

x 3 2x57 2x35x6

Recall that the solution set for an equation is the set of all values for the variable (or ordered pair) that make the equation a true statement. Similarly, the solution set for an in-equality is the set of all values that make the inin-equality a true statement. Example 3 looks at the graphical approach to solving an inequality.

Example 3

Solving an Inequality Graphically

Solve the inequality graphically. 2x57

First, rewrite the inequality as a comparison of two functions. Here, f(x)g(x), in which f(x)2x5 and g(x)7.

NOTEAlgebraic solution: 2x57

2x5575 2x2

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Now graph the two functions on a single set of axes.

Next, draw a vertical dotted line through the point of intersection of the two functions. In this case, there will be a vertical line through the point (1, 7).

The solution set is every xvalue that results inf(x) being greater than g(x), which is every

x value to the right of the dotted line.

Finally, we express the solution set in set notation

xx1 y f x g y f x g y f x g C H E C K Y O U R S E L F 3

Solve the inequality 3x24graphically. NOTEThe solution set will be

all the xvalues that make the original statement, 2x57, true.

Here we ask the question, “For what values of xis the graph of fabove the graph of g?”

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SOLVINGEQUATIONS ANDINEQUALITIESGRAPHICALLY SECTION4.4 249

In Example 3, the function g(x)7 resulted in a horizontal line. In Example 4, we see that the same method works when comparing any two functions.

Example 4

Solving an Inequality Graphically

Solve the inequality graphically. 2x3 5x

First, rewrite the inequality as a comparison of two functions. Here, f(x) g(x), and

f(x)2x3 and g(x)5x.

Now graph the two functions on a single set of axes.

As in Example 3, draw a vertical line through the point of intersection of the two functions. The vertical line will go through the point (1,5). In this case, the line is included (greater than or equal to), so the line is solid, not dotted.

Again, we need to mark every xvalue that makes the statement true. In this case, that is every xfor which the line representing f(x) is above or intersects the line representing

g(x). That is the region in which f(x) is greater than or equal to g(x). We mark the xvalues to the left of the line, but we also want to include the xvalue on the line, so we make it a bracket rather than a parenthesis.

Finally, we express the solutions in set notation. We see that the solution set is every x

value less than or equal to 1, so we write

xx 1 (1, 5) y g f x (1, 5) y g f x NOTEAlgebraic solution

2x3 5x

2x5x3 5x5x 3x3 0

3x33 03

3x 3

(note what happens when we divide by a negative number)

x1

3x 3

3

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The following algorithm summarizes our work in this section.

Solve the inequality graphically. 3x2 2x8

Step 1 Rewrite the inequality as a comparison of two functions.

f(x)g(x) f(x)g(x) f(x)g(x) f(x) g(x) Step 2 Graph the two functions on a single set of axes.

Step 3 Draw a vertical line through the point of intersection of the two graphs. Use a dotted line if equality is not included (or ). Use a solid line if equality is included (or ).

Step 4 Mark the xvalues that make the inequality a true statement. Step 5 Write the solutions in set notation.

Step by Step: Solving an Inequality in One Variable Graphically

The examples we have shown yielded intersections at xvalues that are integers. If the x

value of the intersection is not an integer, it can be very difficult to read from a hand-drawn graph. If a graphing calculator is used, the trace feature can be used to get a very good ap-proximation of the intersection point.

C H E C K Y O U R S E L F A N S W E R S

1. f(x) 3x6 2. f(x) 3x4

g(x)0 g(x)2x1

Solution set 2 Solution set 1

3. 4. y x g f {xx 2} y x g f {xx 2} y g f x (1, 1) y g f (2, 0)

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Exercises

Graphically solve the following equations.

1. 2x80 2. 4x120 3. 7x70 4. 2x60 5. 5x82 6. 4x5 3 7. 2x37 8. 5x94 y x y x y x y x y x y x y x y x

4.4

Name Section Date ANSWERS 1. 2. 3. 4. 5. 6. 7. 8. 251

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Solve the linear equations algebraically, then graphically display the solutions.

9. 3x22x1 10. 4x3 x2 11. 12. 2x33x2 13. 3(x1)4x5 14. 2(x1)5x7 15. 16. 2(2x1) 2x10 y x y x 7

1 5x 1 7

x 1 y x y x y x y x 7 5x 3 2 5x 6 y x y x ANSWERS 9. 10. 11. 12. 13. 14. 15. 16. 252

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In exercises 17 to 32, solve each inequality graphically. 17. 2x8 18. x4 19. 20. 21. 6x 6 22. 3x6 23. 7x7 2x2 24. 7x2x4 y x y x y x y x y x y x 3x 3 4 3 x 3 2 1 y x y x © 2001 McGraw-Hill Companies ANSWERS 17. 18. 19. 20. 21. 22. 23. 24. 253

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© 2001 McGraw-Hill Companies 25. 2x7 3(x1) 26. 2(3x1)4(x1) 27. 6(1x) 2(3x5) 28. 2(x5) 2x1 29. 30. 4x12x8 31. 4x62x2(5x12) 32. 5x32(4x)7x y x y x y x y x 3x 4x 5 3 y x y x y x y x ANSWERS 25. 26. 27. 28. 29. 30. 31. 32. 254

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In exercises 33 to 38, solve the following applications.

33. Business. The cost to produce xunits of wire is C(x)50x5000, and the revenue generated is R(x)60x. Find all values of xfor which the product will at least break even.

34. Business. Find the values of xfor which a product will at least break even if the cost is C(x)85x900 and the revenue is given by R(x)105x.

35. Car Rental. Tom and Jean went to Salem, Massachusetts, for 1 week. They needed to rent a car, so they checked out two rental firms. Wheels, Inc. wanted $28 per day with no mileage fee. Downtown Edsel wanted $98 per week and 14¢ per mile. Set up equations to express the rates of the two firms, and then decide when each deal should be taken.

36. Mileage. A fuel company has a fleet of trucks. The annual operating cost per truck is

C(x)0.58x7800, in which xis the number of miles traveled by a truck per year. What number of miles will yield an operating cost that is less than $25,000?

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Hall. They can spend at the most $3000 for the reception. If the hall charges a $250 cleanup fee plus $25 per person, find the largest number of people they can invite.

38. Tuition. A nearby college charges annual tuition of $6440. Meg makes no more than $1610 per year in her summer job. What is the smallest number of summers that she must work to make enough for 1 year’s tuition?

39. Graphing. Explain to a relative how a graph is helpful in solving each inequality below. Be sure to include the significance of the point at which the lines meet (or what happens if the lines do not meet).

(a) 3x25 (b) 3x24x (c) 4(x1) 24x

40. College. Look at the data here about enrollment in college. Assume that the changes occurred at a constant rate over the years. Make one linear graph for men and one for women, but on the same set of axes. What conclusions could you draw from reading the graph?

No., in Millions, of Men in No., in Millions, of Women in Year the U.S. Enrolled in College the U.S. Enrolled in College

1960 2.3 1.2 1991 6.4 7.8 ANSWERS 37. 38. 39. 40. 256

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Answers

1. 3. 5. 7. 9. 11. 13. 15. y x g(x) x 1 {5} (5, 6) f(x) 7( x ₅1 ₇1) y x f(x) 3(x 1) g(x) 4x 5 {2} (2, 3) y x {5} (5, 4) g(x) ₅2x 6 f(x) ₅7x 3 y x f(x) 3x 2 g(x) 2x 1 {3} (3, 7) y x f(x) 2x 3 {5} g(x) 7 y x f(x) 5x 8 {2} g(x) 2 y x g(x) 0 f(x) 7x 7 {1} y x f(x) 2x 8 g(x) 0 {4} © 2001 McGraw-Hill Companies 257

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17. 19.

21. 23.

25. 27.

29. 31.

33. xx 500 35. If miles are under 700, Downtown Edsel; if over 700, Wheels, Inc.; W$287$196; DE980.14x (xis number of miles)

37. 110 people 39. f(x) 4x 6 g(x) 8x 24 {xx 2.5} (2.5, 4) y x f(x) 3x {xx 1} y x g(x) 3 4x 5 f(x) 6(1 x) g(x) 2(3x 5) {xx R} y x f(x) 2x 7 g(x) 3(x 1) {xx 2} x y f(x) 7x 7 g(x) 2x 2 {xx 1} y x f(x) 6x g(x) 6 {xx 1} y x g(x) 1 {xx 5} f(x) x 3 2 y x f(x) 2x g(x) 8 {xx 4} (4, 8) y x 258