FP2 Chapter 1 - Inequalities Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 3rd August 2015

RECAP ? ? Solve 2 𝑥 2 <𝑥+3 𝑥<2 → 3𝑥<6 ? Solving Quadratic Inequalities (C1) Permitted operations in inequalities? Solve 2 𝑥 2 <𝑥+3 2 𝑥 2 −𝑥−3<0 2𝑥−3 𝑥+1 <0 𝑥<2 → 3𝑥<6 ? 6𝑥<12 → 𝑥<2 ? −6𝑥<12 → 𝑥<−2 ? 1 𝑥 >1 → 1>𝑥 ? ? 𝑦 ? If we divide by a negative number the inequality should ‘flip’. Because in the last example we don’t know whether 𝑥 is positive or negative, we don’t know whether the inequality will flip. In general, never multiply be an expression involving a variable in an inequality unless you know it’s positive. 𝑥 −1 3 2 −1<𝑥< 3 2

Getting around this problem Solve 𝑥 2 𝑥−2 <𝑥+1, 𝑥≠2 We can’t multiply by 𝑥−2, but what could we multiply by? 𝒙−𝟐 𝟐 , which is always positive! Then: 𝑥 2 𝑥−2 < 𝑥+1 𝑥−2 2 𝑥 2 𝑥−2 − 𝑥+1 𝑥−2 2 <0 𝑥−2 𝑥 2 −(𝑥+1)(𝑥−2) <0 𝑥−2 𝑥+2 <0 ? Bro Tip: Remember in FP1 with summation proofs, we tried to avoid expanding things where we could factorise first? ? ? ? ? 𝑦 𝑥 −2 2 −2<𝑥<2

Another Example Solve 𝑥 𝑥+1 ≤ 2 𝑥+3 , 𝑥≠−1, 𝑥≠−3 ? Factorise ? Sketch? Multiply both sides by: 𝑥+1 2 𝑥+3 2 𝑥 𝑥+1 𝑥+3 2 ≤2 𝑥+1 2 𝑥+3 𝑥 𝑥+1 𝑥+3 2 −2 𝑥+1 2 𝑥+3 ≤0 (𝑥+1)(𝑥+3)(𝑥 𝑥+3 −2 𝑥+1 ≤0 𝑥+1 𝑥+3 𝑥 2 −𝑥−2 ≤0 𝑥+1 𝑥+3 𝑥+2 𝑥−1 ≤0 Bro Note: While cubics have two ‘zig zags’ (not necessarily turning points!), quartics have three. If the 𝑥 4 term is positive the quartic starts from the top. Factorise ? Sketch? 𝑦 Solution ? −3≤𝑥≤−2 or −1≤𝑥≤1 𝑥 −3 −2 −1 1

Test Your Understanding FP1 June 2007 Q1 (Yes, it’s moved module!) Find the set of values of 𝑥 for which 𝑥+1 2𝑥−3 < 1 𝑥−3 (7) ? P4 Jan 2006 Q2 (We’re going back to the good ole’ days now of P!) Find the set of values of 𝑥 for which 𝑥 2 𝑥−2 >2𝑥 (6) ?

Exercise 1A Solve the following inequalities. 3 𝑥 2 +5 𝑥+5 >1 −𝟓<𝒙<𝟎 𝒐𝒓 𝒙> 𝟏 𝟑 3𝑥 𝑥−2 >𝑥 𝒙<𝟎 𝒐𝒓 𝟐<𝒙<𝟓 1+𝑥 1−𝑥 > 2−𝑥 2+𝑥 𝒙<−𝟐 𝒐𝒓 𝟎<𝒙<𝟏 𝑥 2 +7𝑥+10 𝑥+1 >2𝑥+7 𝒙<−𝟑 𝒐𝒓 −𝟏<𝒙<𝟏 𝑥+1 𝑥 2 >6 − 𝟏 𝟑 <𝒙< 𝟏 𝟐 𝒃𝒖𝒕 𝒙≠𝟎 𝑥 2 𝑥+1 > 1 6 −𝟏<𝒙<− 𝟏 𝟑 𝒐𝒓 𝒙> 𝟏 𝟐 ? ? 11 𝑥 2 <5𝑥+6 −𝟏<𝒙<𝟔 𝑥 𝑥+1 ≥6 𝒙≥𝟐 𝒐𝒓 𝒙≤−𝟑 2 𝑥 2 +1 >1 −𝟏<𝒙<𝟏 2 𝑥 2 −1 >1 − 𝟑 <𝒙<−𝟏 𝒐𝒓 𝟏<𝒙< 𝟑 𝑥 𝑥−1 ≤2𝑥 𝒙> 𝟑 𝟐 𝒐𝒓 𝟎<𝒙<𝟏 3 𝑥+1 < 2 𝑥 𝒙<−𝟏 𝒐𝒓 𝟎<𝒙<𝟐 3 𝑥+1 𝑥−1 <1 𝒙<−𝟐 𝒐𝒓 −𝟏<𝒙<𝟏 𝒐𝒓 𝒙>𝟐 2 𝑥 2 ≥ 3 𝑥+1 𝑥−2 −𝟏<𝒙<𝟐 𝒙≠𝟎 𝒐𝒓 −𝟏<𝒙<𝟎 𝒐𝒓 𝟎<𝒙<𝟐 2 𝑥−4 <3 𝒙<𝟒 𝒐𝒓 𝒙> 𝟏𝟒 𝟑 3 𝑥+2 > 1 𝑥−5 −𝟐<𝒙<𝟓 𝒐𝒓 𝒙>𝟖.𝟓 1 2 ? ? 12 ? 3 ? 13 ? 4 14 ? 5 ? ? ? 6 15a ? 15b 7 ? 8 ? ? 9 ? 10

Solving by Sketching ? Hence solution: 𝟎<𝒙<𝟏 A simple example: By using appropriate sketches, solve 1 𝑥 >1 𝑦 Click to Brosketch 𝑦= 1 𝑥 1 𝒚=𝟏 𝒚= 𝟏 𝒙 Click to Brosketch 𝑦=1 𝑥 1 The 𝑥 values where the lines intersect (which we call the ‘critical values’) is important. Hence solution: 𝟎<𝒙<𝟏 ?

More Examples On the same axes sketch the graphs of the curves with equations 𝑦= 7𝑥 3𝑥+1 and 𝑦=4−𝑥. Find their points of intersection. Hence solve 7𝑥 3𝑥+1 <4−𝑥 𝑦 As 𝑥→∞, the +1 becomes inconsequential, thus we get 7𝑥 3𝑥 = 7 3 Click to Brosketch 𝑦= 7𝑥 3𝑥+1 𝒚= 𝟕 𝟑 Click to Brosketch 𝑦=4−𝑥 𝑥 − 2 3 2 Points of intersection (by solving as an equation): 𝒙=− 𝟐 𝟑 𝒐𝒓 𝟐 When 𝑥=0, 𝑦=0 𝒙=− 𝟏 𝟑 ? Hence solution to inequality: 𝒙<− 𝟐 𝟑 𝒐𝒓 − 𝟏 𝟑 <𝒙<𝟐 Not defined when 𝑥=− 1 3 ?

Modulus-ey ones ? ? ? ? Solve 𝑥 2 −4𝑥 <3 Solve 3𝑥 +𝑥≤2 𝑦 First isolate modulus to make sketching easier: 3𝑥 ≤2−𝑥 (Noting that 𝑥 2 −4𝑥 =𝑥 𝑥−4 ) 𝒚= 𝒙 𝟐 −𝟒𝒙 𝑦 𝒚=𝟑 2 𝑥 2− 7 1 3 2+ 7 1 2 𝑥 −1 2 Brosketch 𝑦=| 𝑥 2 −4𝑥| Brosketch 𝑦=3 Brosketch 𝑦=|3𝑥| Brosketch 𝑦=2−𝑥 Find critical values: 𝑥 2 −4𝑥=3 → 𝑥=2± 7 − 𝑥 2 −4𝑥 =3 → 𝑥=1 𝑜𝑟 3 Find critical values: 3𝑥=2−𝑥 → 𝑥= 1 2 −3𝑥=2−𝑥 → 𝑥=−1 ? ? Hence solutions: 2− 7 <𝑥<1 𝑜𝑟 3<𝑥<2+ 7 Find critical values: −1≤𝑥≤ 1 2 ? ?

Modulus-ey ones ? ? Solve 𝑥 2 −19 ≤5 𝑥−1 𝑦 Solving 𝑥 2 −19=5 𝑥−1 gives 𝑥=−2 or 𝑥=7 Solving − 𝑥 2 −19 =5 𝑥−1 gives 𝑥=−8 or 3. Which critical values do we actually want? (hint: look at graph) From the positive 𝒙 𝟐 −𝟏𝟗 graph we want the latter point (𝒙=𝟕) For the negative 𝒙 𝟐 −𝟏𝟗 graph we again want the latter point 𝒙=𝟑 Therefore solutions: 𝟑≤𝒙≤𝟕 ? 𝑥 3 7 ?

Test Your Understanding FP1 June 2006 Q7 (a) Use algebra to find the exact solutions of the equation:   2 𝑥 2 +𝑥−6 =6−3𝑥 (6) (b) On the same diagram, sketch the curve with equation y = 2x2 + x – 6 and the line with equation y = 6 – 3x. (3) (c) Find the set of values of x for which 2 𝑥 2 +𝑥−6 >6−3𝑥  (3) (a) ? (c) ? (b) ?

Exercise 1B ? ? ? ? ? ? ? ? ? ? Solve the following inequalities: 9 On the same axes sketch the graphs of 𝑦= 1 𝑥 and 𝑦= 𝑥 𝑥+2 Hence solve 1 𝑥 > 𝑥 𝑥+2 −𝟐<𝒙<−𝟏 𝒐𝒓 𝟎<𝒙<𝟐 On the same axes sketch the graphs of 𝑦= 1 𝑥−𝑎 and 𝑦=4|𝑥−𝑎| Solve 1 𝑥−𝑎 <4 𝑥−𝑎 𝒙<𝒂 𝒐𝒓 𝒙>𝒂+ 𝟏 𝟐 𝑥−6 >6𝑥 𝒙< 𝟔 𝟕 𝑡−3 > 𝑡 2 −𝟏− 𝟏𝟑 𝟐 <𝒕< −𝟏+ 𝟏𝟑 𝟐 𝑥−2 𝑥+6 <9 −𝟕<𝒙<−𝟐− 𝟕 or −𝟐+ 𝟕 <𝒙<𝟑 2𝑥+1 ≥3 𝒙≤−𝟐 𝒐𝒓 𝒙≥𝟏 2𝑥 +𝑥>3 𝒙<−𝟑 𝒐𝒓 𝒙>𝟏 𝑥+3 𝑥 +1 <2 𝒙<− 𝟏 𝟑 𝒐𝒓 𝒙>𝟏 3−𝑥 𝑥 +1 >2 −𝟏<𝒙< 𝟏 𝟑 𝑥 𝑥+2 <1−𝑥 𝒙<−𝟏− 𝟑 𝒐𝒓 − 𝟐 <𝒙<−𝟏+ 𝟑 ? 1 ? 2 ? 10 3 ? 4 ? 5 ? ? ? 6 ? 7 8 ?