In this 2.9 SOLVING INEQUALITIES AND

1 120(2-56)Chapter 2 Linear Equations and INEQUALITIES in One a ski an inequality in the variable xfor the degree measure of the smallest angle of the triangleshown in the figure, given that the degree measure of thesmallest angle is at most 30 .180 x (x 8) 30where Nis the number of teeth on the chainring (by thepedal), nis the number of teeth on the cog (by the wheel),and wis the wheel diameter in inches (Cycling,Burkett andDarst). The following chart gives uses for the various FOR EXERCISE 83x 8x?FIGURE FOR EXERCISE Parcel Service defines the girth ofa box as the sum of the length, twice the width, and twicethe height. The maximum girth that UPS will ship is 130 a box has a length of 45 in.

2 And a width of 30 in., thenwhat inequality must be satisfied by the height?45 2(30) 2h one point during the 1997 season, JayLopez of the Atlanta Braves had 93 hits in 317 times at batfor an average of 93 317 or If he gets xhits in thenext 20 times at bat to get his average over , then whatinequality must xsatisfy? gear gear ratio rfor a bicycle is definedby the formular ,Nw n93 x 317 20 FIGURE FOR EXERCISE 86 RatioUser 90hard pedaling on level ground70 r 90 moderate effort on level ground50 r 70 mild hill climbing35 r 50 long hill climbing with loadSOLVING INEQUALITIES ANDAPPLICATIONSTo solve equations, we write a sequence of equivalent equations that ends in a verysimple equation whose solution is obvious.

3 In this section you will learn that theprocedure for SOLVING INEQUALITIES is the same. However, the rules for performingoperations on each side of an inequality are slightly different from the rules 9In thissection Rules for INEQUALITIES SOLVING INEQUALITIES Applications of InequalitiesA bicycle with a 27-inch diameter wheel has 50 teeth on thechainring and 17 teeth on the cog. Find the gear ratio andindicate what this gear ratio is good , moderate effort on level INEQUALITIES and Applications(2-57)121helpfulhintYou can think of an inequalitylike a seesaw that is out > 20If the same weight is added toor subtracted from each side, itwill remain in the same stateof in the habit of checkingyour work and having confi-dence in your answers.

4 Theanswers to the odd-numberedexercises are in the back of thisbook, but you should look inthe answer section only afteryouhavecheckedonyourown. You will not always havean answer section for InequalitiesEquivalent inequalitiesare INEQUALITIES that have exactly the same such asx> 3 and x 2 5 are equivalent because any number thatis larger than 3 certainly satisfiesx 2 5 and any number that satisfiesx 2 5must certainly be larger than can get equivalent INEQUALITIES by performing operations on each side of aninequality just as we do for SOLVING equations. If we start with the inequality6 10 and add 2 to each side, we get the true statement 8 12.

5 Examine the re-sults of performing the same operation on each side of 6 these operations on each side:Add 2 Subtract 2 Multiply by 2 Divide by 2 Start with 6 108 124 812 203 5 All of the resulting INEQUALITIES are correct. Now if we repeat these operations using 2, we get the following these operations on each side:Add 2 Subtract 2 Multiply by 2 Divide by 2 Start with 6 104 88 12 12 20 3 5 Notice that the direction of the inequality symbol is the same for all of the resultsexcept the last two. When we multiplied each side by 2 and when we divided eachside by 2, we had to reverse the inequality symbol to get a correct result. Thesetables illustrate the rules for SOLVING Property of InequalityIf we add the same number to each side of an inequality we get an equivalentinequality.

6 If a b, then a c b addition property of inequality also allows us to subtract the same number fromeach side of an inequality because subtraction is defined in terms of Property of InequalityIf we multiply each side of an inequality by the same positivenumber, we getan equivalent inequality . Ifa bandc 0, then ac bc. If we multiplyeach side of an inequality by the same negativenumber and reverse the in-equality symbol,we get an equivalent inequality . Ifa bandc 0, thenac multiplication property of inequality also allows us to divide each side of an in-equality by a nonzero number because division is defined in terms of if we multiply or divide each side by a negative number, the inequality symbolis the signs of num-bers, changes their relative po-sition on the number line.

7 Forexample, 3 lies to the left of 5on the number line, but 3 liesto the right of 5. So 3 5,but 3 5. Since multiply-ing and dividing by a negativecause sign changes, these op-erations reverse the 1 Writing equivalent inequalitiesWrite the appropriate inequality symbol in the blank so that the two INEQUALITIES )x 3 9, x_____ 6b) 2x 6, x_____ 3 Solutiona)If we subtract 3 from each side ofx 3 9, we get the equivalent inequal-ityx )If we divide each side of 2x 6 by 2, we get the equivalent inequalityx 3. We use the properties of inequality just as we use the proper-ties of equality. However, when we multiply or divide each side by a negativenumber, we must reverse the inequality InequalitiesTo solve INEQUALITIES , we use the properties of inequality to isolate xon one 2 Using the properties of inequalitySolve and graph the inequality 4x 5 5 19 Original inequality4x 5 5 19 5 Add 5 to each 6 Divide each side by the last inequality is equivalent to the first, they have the same solutions andthe same graph, which is shown in Fig.

8 In the next example we divide each side of an inequality by a negative can use the TABLE feature of a graph-ing calculator to numerically support thesolution to the inequality 4x 5 19 inExample 2. Use the Y = key to enter theequation y1 4x , use TBLSET to set the table so that thevalues of xstart at and the change in xis that whenxis larger than 6,y1(or 4x 5) is larger than 19. Note that thistable is not a method for SOLVING an in-equality, it is merely a way of verifying orsupporting the algebraic close-up45678123 FIGURE , press TABLE to see lists of x-valuesand the corresponding (2-58)Chapter 2 Linear Equations and INEQUALITIES in One INEQUALITIES and Applications(2-59)123 EXAMPLE 3 Reversing the inequality symbolSolve and graph the inequality 5 5x 1 2(5 x).

9 Solution5 5x 1 2(5 x)Original inequality5 5x 11 2xSimplify the right 3x 11 Add 2xto each side. 3x 6 Subtract 5 from each 2 Divide each side by 3, and reverse the INEQUALITIES 5 5x 1 2(5 x) andx 2 have the same graph, whichis shown in Fig. We can use the rules for SOLVING INEQUALITIES on the compound INEQUALITIES thatwe studied in Section 4 SOLVING a compound inequalitySolve and graph the inequality 9 23x 7 9 23x 7 5 Original inequality 9 7 23x 7 7 5 7 Add 7 to each part. 2 23x 12 Simplify. 32 ( 2) 32 23x 32 12 Multiply each part by 32 . 3 x number that satisfies 3 x 18 also satisfies the original shows all of the solutions to the original inequality .

10 There are many negative numbers in Example 4, but theinequality was not reversed, since we did not multiply or divide by a negative num-ber. An inequality is reversed only if you multiply or divide by a negative 5 Reversing inequality symbols in a compound inequalitySolve and graph the inequality 3 5 x 3 5 x 5 Original inequality 3 5 5 x 5 5 5 Subtract 5 from each part. 8 x 0 Simplify.( 1)( 8) ( 1)( x) ( 1)(0)Multiply each part by 1, reversingthe inequality x 0 FIGURE 6 5 4 301 2 10369 12 15 18 3 FIGURE (2-60)Chapter 2 Linear Equations and INEQUALITIES in One VariableIt is customary to write 8 x 0 with the smallest number on the left:0 x 8 Figure shows all numbers that satisfy 3 5 x 5.