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Year 9 Inequalities Dr J Frost ([email protected]) Objectives: Solving linear inequalities, combining inequalities and representing solutions on number lines. Last modified: 23rd March 2015 Writing inequalities and drawing number lines You need to be able to sketch equalities and strict inequalities on a number line. This is known as a βstrictβ inequality. x>3 Means: x is (strictly) greater ? than 3. 0 1 2 3 4 x < -1 Means: x is (strictly) less?than -1. 5 -3 -2 -1 ? 4 5 ? 2 xβ€5 Means: x is greater than?or equal to 4. 3 1 ? xβ₯4 2 0 6 7 Means: x is less than or ? equal to 4. 2 3 4 5 ? 6 7 Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed? π>π Can we add or subtract to both sides? πβπ>π Click to οΌ Deal Click to ο» No Deal Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed? ππ > π π>π Click to οΌ Deal Can we divide both sides by a positive number? Click to ο» No Deal Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed? π<π Can we multiply both sides by a positive number? ππ < π Click to οΌ Deal Click to ο» No Deal Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed? π<π Can we multiply both sides by a negative number? βπ < βπ Click to ο»Deal Click toοΌ No Deal βFlippingβ the inequality If we multiply or divide both sides of the inequality by a negative number, the inequality βflipsβ! OMG magic! -2 2 < -4 4 Click to start Bro-manimation Alternative Approach Or you could simply avoid dividing by a negative number at all by moving the variable to the side that is positive. βπ₯ < 3 ? π₯ β3 < π₯ > ?β3 1 β 3π₯ β₯ 7 1 β 7 ?β₯ 3π₯ β6 β₯ ?3π₯ β2 β₯ ?π₯ ? π₯ β€ β2 Quickfire Examples 2π₯ < 4 βπ₯ > β3 4π₯ β₯ 12 β4π₯ > 4 π₯ β β€1 2 Solve Solve Solve Solve Solve π₯ <? 2 π₯ <? 3 π₯ β₯? 3 π₯ <?β1 π₯ β₯?β2 Deal or No Deal? We can manipulate inequalities in various ways, but which of these are allowed and not allowed? 1 <2 π₯ Can we multiply both sides by a variable? 1 < 2π₯ Click to ο»Deal Click toοΌ No Deal The problem is, we donβt know if the variable has a positive or negative value, so negative solutions would flip it and positive ones wouldnβt. You wonβt have to solve questions like this until Further Maths A Level! More Examples 3π₯ β 4 < 20 4π₯ + 7 > 35 π₯ 5 + β₯ β2 2 7 β 3π₯ > 4 π₯ 6β β€1 3 Hint: Do the addition/subtraction before you do the multiplication/division. Solve π₯< ? 8 Solve Solve Solve Solve π₯ >? 7 π₯ β₯ ?β14 π₯ <? 1 π₯ β₯? 15 Dealing with multiple Hint: inequalities Do the addition/subtraction before you do the multiplication/division. 8 < 5x 5x -- 22 β€ 23 and 2 < x and x β€ 5 π<πβ€π Click to start bromanimation More Examples π < ππ + π < π βπ < βπ < π Hint: Do the addition/subtraction before you do the multiplication/division. Solve Solve βπ < ?π < π βπ < ?π < π Test Your Understanding ππ < ππ β π < ππ π < π β ππ < π Solve Solve π < π? < π βπ < ?π < π Exercise 1 Solve the following inequalities, and illustrate each on a number line: 1 2 3 4 5 6 7 8 9 10 11 N1 2π₯ β 1 > 5 π >?π β2π₯ < 4 π >?π 5π₯ β 2 β€ 3π₯ + 4 π β€?π N2 π₯ +1β₯6 π β₯?ππ 4 π¦ β1β€7 π β€?ππ 6 1βπ¦ π β€π¦ π β₯? 2 π 1 β 4π₯ > 5 π <?βπ 5 β€ 2π₯ β 1 < 9 π β€ ?π < π 5 β€ 1 β 2π₯ < 9 β π < ?π β€ βπ 10 + π₯ < 4π₯ + 1 < 33 π < π? < π 1 β 3π₯ < 2 β 2π₯ < 3 β π₯ π >?βπ Sketch the graphs for 1 π¦ = π₯ and π¦ = 1. 1 Hence solve π₯ > 1 0<x<1 ? You can get around the problem of multiplying/dividing both sides by an expression involving a variable, by separately considering when itβs positive, and when itβs negative, and putting this together. Hence solve: 3 >4 π₯+2 If we assume π + π is positive, then π > π β π and solving gives π < β . Thus βπ < π π π < β as we had to assume π > βπ. If ? π π π < βπ then this solves to π > β which π is a contradiction. Thus βπ < π < β π π Combining inequalities Itβs absolutely crucial that you distinguish between the words βandβ and βorβ when constraining the values of a variable. AND How would we express βx is greater than or equal to 2, and less than 4β? ? x<4 x β₯ 2 and x β₯ 2,?x < 4 2 β€ x? < 4 This last one emphasises the fact that x is between 2 and 4. OR How would we express βx is less than -1, or greater than 3β? ? x>3 x < -1 or This is the only way you would write this β you must use the word βorβ. Combining inequalities Itβs absolutely crucial that you distinguish between the words βandβ and βorβ when constraining the values of a variable. 2β€x<4 0 1 2 3 ? x < -1 or x > 4 4 5 -1 0 1 2 ? 3 4 Combining inequalities Itβs absolutely crucial that you distinguish between the words βandβ and βorβ when constraining the values of a variable. To illustrate the difference, what happens when we switch them? or and x β₯ 2 and x < 4 0 1 2 3 ? 4 x < -1 or x > 4 5 -1 0 1 2 ? 3 4 I will shoot you if I see any of theseβ¦ 4>π₯<8 This is technically equivalent to: x<4 ? 4<π₯>7 This is technically equivalent to: x>7 ? 7>π₯>4 The least offensive of the three, but should be written: 4<x<7 ? Combining Inequalities In general, we can combine inequalities either by common sense, or using number lines... 2 5 Where are you on both lines? 4 Combined ? 2 2<π₯<5 π>π 5 4 π₯<4 Combined ?π < π < π Test Your Understanding ? 1st 2nd -1 condition condition Combined -3 ? 3 ? ? 5 Exercise 2 By sketching the number lines or otherwise, combine the following inequalities. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?