1. FP2 Chapter 1 - Inequalities
Dr J Frost
Last modified: 3rd August 2015

2. 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< ?
6𝑥<12 → 𝑥< ?
−6𝑥<12 → 𝑥<−2 ?
1 𝑥 > → 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

3. 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− 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

4. Another Example Solve 𝑥 𝑥+1 ≤ 2 𝑥+3 , 𝑥≠−1, 𝑥≠−3 ? Factorise ? Sketch?
Multiply both sides by: 𝑥 𝑥+3 2
𝑥 𝑥+1 𝑥+3 2 ≤2 𝑥 𝑥+3 𝑥 𝑥+1 𝑥+3 2 −2 𝑥 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

5. 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) ?

6. Exercise 1A
Solve the following inequalities.
3 𝑥 2 +5 𝑥+5 > −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 > − 𝟏 𝟑 <𝒙< 𝟏 𝟐 𝒃𝒖𝒕 𝒙≠𝟎 𝑥 2 𝑥+1 > −𝟏<𝒙<− 𝟏 𝟑 𝒐𝒓 𝒙> 𝟏 𝟐 ? ? 11
𝑥 2 <5𝑥 −𝟏<𝒙<𝟔 𝑥 𝑥+1 ≥ 𝒙≥𝟐 𝒐𝒓 𝒙≤−𝟑 2 𝑥 2 +1 > −𝟏<𝒙<𝟏 2 𝑥 2 −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 < 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 < 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

7. 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 ?

8. 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 ?

9. 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= → 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 ? ?

10. 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 ?

11. 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) ?

12. 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 −𝟏− 𝟏𝟑 𝟐 <𝒕< −𝟏+ 𝟏𝟑 𝟐
𝑥−2 𝑥+6 < −𝟕<𝒙<−𝟐− 𝟕 or −𝟐+ 𝟕 <𝒙<𝟑
2𝑥+1 ≥ 𝒙≤−𝟐 𝒐𝒓 𝒙≥𝟏
2𝑥 +𝑥> 𝒙<−𝟑 𝒐𝒓 𝒙>𝟏
𝑥+3 𝑥 +1 < 𝒙<− 𝟏 𝟑 𝒐𝒓 𝒙>𝟏
3−𝑥 𝑥 +1 > −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 ?