Inequalities. After completing this chapter you should be able to:
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1 After completing this chapter ou should be able to: Manipulate inequalities Determine the critical values of an inequalit Find solutions of algebraic inequalities Inequalities 1 Most applications of mathematics require the solution of inequalities at some stage In manufacturing or business ou will want to know what level of price will ensure that our profit is greater than our production costs this means solving inequalities 1
2 CHAPTR You can manipulate inequalities to solve them In C1 ou learnt how to solve simple quadratic inequalities b rearranging them. The inequalit sign can be treated like an equals sign as long as ou do not divide or multipl both sides of the epression b a negative number. There are three steps to solving inequalities. ample 1 Solve 0 0 So ( )( 1) 0 So the critical values are _ or 1 A sketch of gives Step 1 is to find the critical values. Rearrange the epression and then replace the inequalit smbol with an equals sign and solve So the solution to 0 is when Step is to draw a sketch, or use a table of values to determine which sets of values satisf the inequalit. Step is to write down the answer b using the graph to interpret the inequalit. In FP ou will dealing with algebraic fractions, and care must be taken when rearranging the inequalit to make sure that ou are not multipling b a quantit that could be negative. ample Solve the inequalit 1, Multipl both sides b ( ) ( ), ( ) ( 1) A natural first step would be to multipl both sides b ( ) but we cannot be sure that this is positive. A simple solution is to multipl both sides of the inequalit b ( ) as this will alwas be positive.
3 Inequalities ( ) ( ) ( ) ( 1) ( ) ( 1)( ) 0 ( )( ( 1)( ) 0 ( )( ) 0 or ( )( ) < 0 Critical values The sketch of ( )( ) is Do not aim to multipl out but cancel, collect terms on one side and factorise. Now the problem is similar to those seen in C1. You find the critical values, draw a sketch and write down the answers. The solution to ( )( ) 0 is The same approach can be used in more complicated situations. ample Solve the inequalit 1 1, This time multipl both sides b ( 1) ( ) So ( 1) ( ) ( 1) ( ) ( 1) ( ) ( 1)( ) ( 1) ( ) 0 ( 1)( )(( ) ( 1)) 0 ( 1)( )( ) 0 ( 1)( )( )( 1) 0 So the critical values are: 1,, or 1 In order to remove the fractions and guarantee that ou are not multipling b a negative quantit, use ( 1) ( ) Cancel terms on each side. Collect terms on LHS To find the critical values ou need to solve. ( 1)( )( )( 1) 0
4 CHAPTR 1 A sketch of ( 1)( )( )( 1) is 1 1 The curve ( 1)( )( ) ( 1) is essentiall an curve, so it starts in top left and ends in top right and passes through 1,, or 1. The eact shape does not matter. So the solution to ( 1)( )( )( 1) 0 is or 1 1 If the question has a as opposed to a then ou must check carefull at the end whether the ends of the set of values are valid. Since 1 or these values must not be included, hence not is used. ercise 1A Solve the following inequalities ( 1) ( 1)( 1) ( 1)( ) a 1 6 b You can use graphs to solve ineualities ample a n the same aes sketch the graphs of the curves with equations b Find the points of intersection of 7 and. 1 c Solve and 1
5 Inequalities a is a straight line crossing the aes at (, 0) and (0, ). 7 has a root at (0, 0) 1 There is a vertical asmptote at _ 1 There is a horizontal asmptote at _ 7 So the sketch looks like this For the sketches look for: 1 Intersections with the aes An vertical asmptotes An horizontal asmptotes to find these Writing 7_ ( might help. ) 1 7_ 1 1 ( ) b ( )( ) 0 So _ or Multipl both sides b 1 Multipl out and collect terms to form a quadratic equation. Solve the equation in the usual wa this one factorises c Marking these points on the graph 7 Look on the sketch for the places where the line is above the curve. These places will give the solution. 1 So the solution is _ or _ 1 Notice how the vertical asmptote has an influence on the second part of the answer. 5
6 CHAPTR 1 The sketching approach is particularl useful if the inequalit involves the modulus function. ample 5 Solve A sketch of and looks like: Since there is a modulus function sketch and on the same aes. To find the critical values, remember that there are two cases to consider when solving. To find the critical values, solve ( ) 0 ( )( 1) 0 1 or Sometimes the quadratic formula ma be required You need to identif where the points of intersection are on the sketch. Marking these values on the sketch: 1 So the solution is: 7 1 or 7 Finall write down the solution to the inequalit the points where the line is above the curve. 6
7 Inequalities Sometimes a little simple rearranging first can make the sketching much simpler ample 6 Solve Rearranging gives: Sketching and gives Sketching is quite difficult so it is usuall simpler to rearrange and isolate the modulus function. Critical values are given b: _ 1 R 1 So the line is above for 1 _ 1 Find the critical values in the usual wa. Remember the two cases. B considering the positions of the critical values, identif the places where the line is above the V shaped graph. Sometimes care must be taken to identif the correct roots when solving the modulus equations. ample 7 Solve the inequalit 19 5( 1) Sketching both graphs First sketch the graphs 7 7
8 CHAPTR ( 7)( ) 0 7 or Then find the critical values. ( 19) ( 8)( ) 0 8 or The line is above the curve when 7 Solving the equations gives four values but the graphs onl have two crossing points. So the valid critical values are and 7. Finall write down the solution ercise 1B Solve the following inequalities: t t ( )( 6) a n the same aes sketch the graphs of 1 and. b Solve 1 10 a n the same aes sketch the graphs of 1 a and a b Solve, giving our answers in terms of the constant a, 1 a, a. Mied eercise 1C 1 Solve the inequalit 7 ( 1) Solve the inequalit 6 1 Find the set of value for which Find the complete set of values of for which 5 Find the set of values of for which 1 1 ( )( 9) 6 Solve 5 1 8
9 Inequalities 7 a Sketch, on the same aes, the graph with equation, and the line with equation 5 1 b Solve the inequalit a Use algebra to find the eact solution of 6 6 b n the same diagram, sketch the curve with equation 6 and the line with equation 6 c Find the set of values of for which a n the same diagram, sketch the graphs of and 1, showing the coordinates of the points where the graphs meet the -ais. b Solve 1, giving our answers in surd form where appropriate. c Hence, or otherwise, find the set of values of for which 1 9
10 CHAPTR 1 Summar of ke points 1 Remember the three steps: find the critical values use a sketch or table to identif the solutions write down the answers checking carefull for validit if using. Tr rearranging the inequalit to make the sketching easier. Remember ou should onl multipl b a quantit which is alwas positive. A sketch is usuall the best approach if a modulus function is involved. 10
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