Notes and Eamples Core Inequalities and indices Section : Errors and inequalities These notes contain subsections on Inequalities Linear inequalities Quadratic inequalities This is an eample resource from MEI s Integral online mathematics resources which cover A level Mathematics and Further Mathematics for AQA, Edecel, OCR, OCR(MEI) and WJEC. As well as notes, eercises, online tests and interactive resources, Integral contains many resources and activities to facilitate and encourage classroom discussion and group work. Inequalities more samples and further details Inequalities are similar to equations, but instead of an equals sign, =, they involve one of these signs: < less than > greater than less than or equal to greater than or equal to This means that whereas the solution of an equation is a specific value, or two or more specific values, the solution of an inequality is a range of values. Inequalities can be solved in a similar way to equations, but you do have to be very careful, as in some situations you need to reverse the inequality. This is shown in these eamples. Linear inequalities A linear inequality involves only terms in and constant terms. Eample Solve the inequality 3 + > Solution You can treat this just like a linear equation. 3 + > + > 5 > 6 > 3 Subtract from each side Subtract from each side Divide both sides by The net eample involves a situation where you have to divide by a negative number. When you are solving an equation, multiplying or dividing by a negative number is not a problem. However, things are different with inequalities. of 6 07/0/ MEI
The statement -3 < is clearly true. If you add something to each side, it is still true If you subtract something from each side, it is still true -3 + < + - < 3-3 < -6 < 4-3 4 < 4-7 < - If you multiply or divide each side by a positive number, it is still true However, if you multiply each side by a negative number then things go wrong! -3 - < - 6 < -4 When you multiply or divide each side by a negative number, you must reverse the inequality. The following eample demonstrates this. Two solutions are given: in the first the inequality is reversed when dividing by a negative number, in the second this situation is avoided by a different approach. Eample Solve the inequality Solution () 3 5 3 6 Subtract from each side Subtract from each side Divide both sides by 3, reversing the inequality. Solution () 3 6 3 Add to each side Add 5 to each side Divide both sides by 3 Finish by writing the inequality the other way round. of 6 07/0/ MEI
You can check that you have the sign the right way round by picking a number within the range of the solution, and checking that it satisfies the original inequality. In the above eample, you could try =. In the original inequality you get 0-3, which is correct. When solving inequalities, make sure that you use the correct inequality sign. If the question involves < or >, the solution will also involve < or > (not necessarily the same one of these as in the question though!). If the question involves or, the solution will also involve one of these. You could waste a mark, for eample by using < when it should be. The Inequalities Activity takes you through the ideas behind some of the different methods of solving linear inequalities, including thinking about graphs. You can look at some more eamples using the Flash resource Linear inequalities. For more practice in solving linear inequalities, try the interactive questions Solving linear inequalities. There is also an Inequalities puzzle, in which you need to cut out all the pieces and match linear inequalities with their solutions to form a large heagon. Quadratic inequalities You can solve a quadratic inequality by factorising the quadratic epression, just as you do to solve a quadratic equation. This tells you the boundaries of the solutions. There are then several possible approaches to finding a solution: sketching a graph using a number line using a table All three of these approaches are shown in Eamples 3 and 4 below. The number line and the table deal with the same idea (looking at the sign of each factor), but they are set out differently. In eaminations, many candidates have difficulty giving the solution to a quadratic inequality, even after factorising correctly. Make sure that you choose an approach that you are comfortable with and get plenty of practice. 3 of 6 07/0/ MEI
Eample 3 Solve the inequality Solution using a graph 6 < 0 ( + )( 3) < 0 8 6 4 6 < 0 6 4 4 6 4 6 8 y This shows that the graph of y = 6 cuts the -ais at = - and = 3. Use this information to sketch the graph. The solution to the inequality is the negative part of the graph. This is the part between and 3. The solution is < < 3. Solution using a number line 6 < 0 ( + )( 3) < 0 Open circles indicate that this point is not included The factor + is positive if > -. The factor 3 is positive if > 3. -3 - - 0 3 4 In this case we need the product of the two factors to be negative. This is the case if one factor only is positive. From the number line, this is when - < < 3. Solution 3 using a table 6 < 0 ( + )( 3) < 0 These are the critical points the points at which the sign may change. The left-hand side is zero when = - and when = 3. < - - < < 3 > 3 + + + 3 + ( + )( 3) + + From the table, the left-hand side of the inequality is negative for - < < 3. The table looks at the sign of each factor for each possible set of values. In the net eample, the term in ² is negative. It is usually best to start by multiplying through by - and changing the inequality sign, as it is easier to work with a positive ² term. 4 of 6 07/0/ MEI
Eample 4 Solve the inequality 3 Solution 3 + 5 3 0 ( + 3)( ) 0 8 y 6 4 4 3 3 4 4 6 8 The solution is -3 or ½. This shows that the graph of y = -3 and = ½. You can now sketch the graph. = + 5 3cuts the -ais at The solution to the inequality is the positive part of the graph. This is in fact two separate parts. Solution using a number line 3 + 5 3 0 ( + 3)( ) 0 The factor + 3 is positive if 3. The factor is positive if. -3 - - 0 3 4 In this case we need the product of the two factors to be positive. This is the case if either both factors are positive or both factors are negative. From the number line, this is when 3 or. Solution 3 using a table 3 + 5 3 0 ( + 3)( ) 0 The left-hand side is zero when = -3 and when =. < -3-3 < < > + 3 + + + ( + 3)( ) + + The table looks at the sign of each factor for each possible set of values. From the table, the left-hand side of the inequality is positive when 3 or. 5 of 6 07/0/ MEI
When there are two parts to the solution, as in Eample 4, it should be written as two separate inequalities. You cannot write 3 or 3 - these suggest that the solution is values of which are less that -3 AND greater than - there are no such values of! Also, as for linear inequalities, make sure that you use the appropriate inequality sign, according to the question. To see more eamples, use the Flash resource Quadratic inequalities. (This uses the number line approach). You can also look at the Solving inequalities video, which uses a range of approaches. For more practice in solving quadratic inequalities, try the interactive questions Solving quadratic inequalities. There is also a Quadratic inequalities puzzle, in which you need to cut out all the pieces and match quadratic inequalities with their solutions to form a large triangle. 6 of 6 07/0/ MEI