Solving Equations and Inequalities Graphically

Transcription

1 4.4 Solvin Equations and Inequalities Graphicall 4.4 OBJECTIVES 1. Solve linear equations raphicall 2. Solve linear inequalities raphicall In Chapter 2, we solved linear equations and inequalities. In this section, we will raphicall demonstrate solutions or similar statements. In usin this section, note that each raphical demonstration is accompanied b the alebraic solution, which appears in the marin. The techniques o this section are not desined as an alternative to the alebra. The are rather an introduction to the idea o viewin a solution. This is a skill that will be ver useul as ou continue to stud mathematics. In our irst eample, we will solve a simple linear equation. The raphical method ma seem cumbersome, but once ou master it, ou will ind it quite helpul, particularl i ou are a visual learner. Eample 1 A Graphical Approach to Solvin a Linear Equation Graphicall solve the ollowin equation. NOTE Alebraicall Step 1 Let each side o the equation represent a unction o. () 2 6 () 0 Step 2 Graph the two unctions on the same set o aes. NOTE We ask the question, when is the raph o equal to the raph o? Speciicall, or what values o does this occur? Step 3 Find the intersection o the two raphs. To do this, eamine the two raphs closel to see where the intersect. Identi the coordinates o that point. This intersection determines the solution to the oriinal equation. 245

2 246 CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS (3, 0) The two lines intersect on the ais at the point (3, 0). We are lookin or the value at the point o intersection, which is 3. CHECK YOURSELF 1 Graphicall solve the ollowin equation The raph o the equation is oten used to check the alebraic solution. This concept is illustrated in Eample 2. Eample 2 Solvin Linear Equations Alebraicall and Graphicall Solve the linear equation alebraicall, then raphicall displa the solution. NOTE Alebraicall The solution set is 2. 2( 3) 3 4 To raphicall displa the solution, let () 2( 3) () 3 4 Graphin both lines, we et

3 SOLVING EQUATIONS AND INEQUALITIES GRAPHICALLY SECTION The point o intersection appears to be ( 2, 2), which conirms that 2 is a reasonable solution to the equation 2( 3) 3 4 CHECK YOURSELF 2 First solve the linear equation alebraicall, then raphicall displa the solution The ollowin alorithm summarizes our work in raphicall solvin an equation. Step b Step: Graphicall Solvin an Equation Step 1 Let each side o the equation represent a unction o. Step 2 Graph the two unctions on the same set o aes. Step 3 Find the intersection o the two raphs. The value at this intersection represents the solution to the oriinal equation. We will now use the raphs o linear unctions to determine the solutions o a linear inequalit. Linear inequalities in one variable,, are obtained rom linear equations b replacin the smbol or equalit ( ) with one o the inequalit smbols (,,, ). The eneral orm or a linear inequalit in one variable is a in which the smbol can be replaced with,, or. Eamples o linear inequalities in one variable include Recall that the solution set or an equation is the set o all values or the variable (or ordered pair) that make the equation a true statement. Similarl, the solution set or an inequalit is the set o all values that make the inequalit a true statement. Eample 3 looks at the raphical approach to solvin an inequalit. NOTE Alebraic solution: Eample 3 Solvin an Inequalit Graphicall Solve the inequalit raphicall First, rewrite the inequalit as a comparison o two unctions. Here, () (), in which () 2 5 and () 7.

4 248 CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS Now raph the two unctions on a sinle set o aes. Here we ask the question, For what values o is the raph o above the raph o? Net, draw a vertical dotted line throuh the point o intersection o the two unctions. In this case, there will be a vertical line throuh the point (1, 7). The solution set is ever value that results in () bein reater than (), which is ever value to the riht o the dotted line. NOTE The solution set will be all the values that make the oriinal statement, 2 5 7, true. Finall, we epress the solution set in set notation 1 CHECK YOURSELF 3 Solve the inequalit raphicall.

5 SOLVING EQUATIONS AND INEQUALITIES GRAPHICALLY SECTION In Eample 3, the unction () 7 resulted in a horizontal line. In Eample 4, we see that the same method works when comparin an two unctions. Eample 4 Solvin an Inequalit Graphicall NOTE Alebraic solution (note what happens when we divide b a neative number) 1 Solve the inequalit raphicall First, rewrite the inequalit as a comparison o two unctions. Here, () (), and () 2 3 and () 5. Now raph the two unctions on a sinle set o aes. ( 1, 5) As in Eample 3, draw a vertical line throuh the point o intersection o the two unctions. The vertical line will o throuh the point ( 1, 5). In this case, the line is included (reater than or equal to), so the line is solid, not dotted. Aain, we need to mark ever value that makes the statement true. In this case, that is ever or which the line representin () is above or intersects the line representin (). That is the reion in which () is reater than or equal to (). We mark the values to the let o the line, but we also want to include the value on the line, so we make it a bracket rather than a parenthesis. ( 1, 5) Finall, we epress the solutions in set notation. We see that the solution set is ever value less than or equal to 1, so we write 1

6 250 CHAPTER 4 GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS CHECK YOURSELF 4 Solve the inequalit raphicall The ollowin alorithm summarizes our work in this section. Step b Step: Solvin an Inequalit in One Variable Graphicall Step 1 Rewrite the inequalit as a comparison o two unctions. () () () () () () () () Step 2 Step 3 Step 4 Step 5 Graph the two unctions on a sinle set o aes. Draw a vertical line throuh the point o intersection o the two raphs. Use a dotted line i equalit is not included ( or ). Use a solid line i equalit is included ( or ). Mark the values that make the inequalit a true statement. Write the solutions in set notation. The eamples we have shown ielded intersections at values that are inteers. I the value o the intersection is not an inteer, it can be ver diicult to read rom a hand-drawn raph. I a raphin calculator is used, the trace eature can be used to et a ver ood approimation o the intersection point. CHECK YOURSELF ANSWERS 1. () () 3 4 () 0 () 2 1 (2, 0) (1, 1) Solution set 2 Solution set { 2} { 2}

7 Name 4.4 Eercises Section Date Graphicall solve the ollowin equations ANSWERS

8 ANSWERS Solve the linear equations alebraicall, then raphicall displa the solutions ( 1) ( 1) (2 1)

9 ANSWERS In eercises 17 to 32, solve each inequalit raphicall

10 ANSWERS ( 1) 26. 2(3 1) 4( 1) (1 ) 2(3 5) 28. 2( 5) (5 12) (4 ) 7 254

11 ANSWERS In eercises 33 to 38, solve the ollowin applications. 33. Business. The cost to produce units o wire is C() , and the revenue enerated is R() 60. Find all values o or which the product will at least break even Business. Find the values o or which a product will at least break even i the cost is C() and the revenue is iven b R() Car Rental. Tom and Jean went to Salem, Massachusetts, or 1 week. The needed to rent a car, so the checked out two rental irms. Wheels, Inc. wanted $28 per da with no mileae ee. Downtown Edsel wanted $98 per week and 14 per mile. Set up equations to epress the rates o the two irms, and then decide when each deal should be taken. 36. Mileae. A uel compan has a leet o trucks. The annual operatin cost per truck is C() , in which is the number o miles traveled b a truck per ear. What number o miles will ield an operatin cost that is less than $25,000? 255

12 ANSWERS Weddin. Eileen and Tom are havin their weddin reception at the Warrinton Fire Hall. The can spend at the most $3000 or the reception. I the hall chares a $250 cleanup ee plus $25 per person, ind the larest number o people the can invite Tuition. A nearb collee chares annual tuition o $6440. Me makes no more than $1610 per ear in her summer job. What is the smallest number o summers that she must work to make enouh or 1 ear s tuition? 39. Graphin. Eplain to a relative how a raph is helpul in solvin each inequalit below. Be sure to include the siniicance o the point at which the lines meet (or what happens i the lines do not meet). (a) (b) (c) 4( 1) Collee. Look at the data here about enrollment in collee. Assume that the chanes occurred at a constant rate over the ears. Make one linear raph or men and one or women, but on the same set o aes. What conclusions could ou draw rom readin the raph? No., in Millions, o Men in No., in Millions, o Women in Year the U.S. Enrolled in Collee the U.S. Enrolled in Collee

13 Answers 1. () {4} () 7 7 {1} () 0 () 0 5. () () 2 3 {2} () 7 () 2 {5} 9. () () () 5 3 (3, 7) {3} (5, 4) {5} () () 3( 1) 15. {2} (2, 3) () () 7( ) 5 7 {5} (5, 6) () 1 257

14 17. () (4, 8) () 8 () 3 2 () 1 { 4} { 5} 21. () { 1} () 6 () 7 7 { 1} () () () 6(1 ) { 2} { R} () 3( 1) () 2(3 5) 29. () () 4 6 (2.5, 4) { 2.5} { 1} 4 5 () 3 () I miles are under 700, Downtown Edsel; i over 700, Wheels, Inc.; W $28 7 $196; DE ( is number o miles) people