1. GCSE: Quadratic Inequalities
Dr J Frost
Last modified: 7th May 2017

2. Reminder: Inequalities
We want to solve 2𝑥+1>5
How many solutions are there to this inequality?
Infinitely many! 𝑥 could be 3, 8, 4.29, etc.
Bro Side Notes: Equations and inequalities are not limited to just one solution: we have seen that quadratics equations often have 2 solutions, and in this example it would not be possible to ‘list’ all the solutions because there are infinitely many! ?
How do we represent the set of all solutions to this inequality?
𝑥>2
i.e. “any real number greater than 2” ?
In this particular example, known as a linear inequality, the range of solutions was simple. But for quadratic inequalities, this might be more complex…

3. Quadratic Inequalities
By trial and error (or otherwise!), try and think of the range (or ranges) of values that satisfies the following inequality.
Your values need not be whole numbers.
𝑥 2 +2𝑥−15>0 ?
Anything above 3 or below -5 works, e.g. when 𝑥 is 4: −15=9 and 9>0
𝑥<−5 𝑜𝑟 𝑥>3
You may have spotted that the -5 and 3 are solutions to the equation 𝑥 2 +2𝑥−15=0.
How then can we get the inequality from these values without guess work?

4. Click to Bro-Bolden >
Solving Quadratic Inequalities
Solve 𝑥 2 +2𝑥−15>0
Step 1: Get 0 on one side
(already done!)
𝑥+5 𝑥−3 >0 ?
Step 2: Factorise
Step 3: Sketch and reason 𝑥 𝑦 −5 3
𝑦=(𝑥+5)(𝑥−3)
Since we sketched 𝑦=(𝑥+5)(𝑥−3)
we’re interested where 𝑦>0, i.e. the parts of the line where the 𝑦 value is positive.
Click to Bro-Bolden >
What can you say about the 𝑥 values of points in this region?
𝒙<−𝟓
What can you say about the 𝑥 values of points in this region?
𝒙>𝟑 ?
𝑥<−5 or 𝑥>3
Bro Note: If the 𝑦 value is ‘strictly’ greater than 0, i.e. > 0, then the 𝑥 value is strictly less than -5. So the < vs ≤ must match the original question. ? ?

5. ? Sketch with highlighted region
Solving Quadratic Inequalities
Solve 𝑥 2 +2𝑥−15≤0
Step 1: Get 0 on one side
(already done!)
𝑥+5 𝑥−3 ≤0
Step 2: Factorise
Step 3: Sketch and reason 𝑥 𝑦 −5 3
𝑦=(𝑥+5)(𝑥−3)
? Sketch with highlighted region
? Final solution
−5≤𝑥≤3
Again, what can we say about the 𝑥 value of any point in this region?
Bro Note: We could write “𝑥≥−5 and 𝑥≤3” but the above better expresses that 𝑥 is “between” -5 and 3.
Bro Note: As discussed previously, we need ≤ rather than < to be consistent with the original inequality.

6. Further Examples Solve 𝒙 𝟐 +𝟓𝒙≥−𝟒 𝑥 2 +5𝑥+4≥0 𝑥+4 𝑥+1 ≥0
𝑥 2 +5𝑥+4≥0 𝑥+4 𝑥+1 ≥0
Solve 𝒙 𝟐 <𝟗 𝑥 2 −9<0 𝑥+3 𝑥−3 <0 ? ? 𝑥 𝑦 −4 −1
𝑦=(𝑥+4)(𝑥+1) 𝑥 𝑦 −3 3
𝑦=(𝑥+3)(𝑥−3)
𝑥≤−4 or 𝑥≥−1
−3<𝑥<3
Bro Note: The most common error I’ve seen students make with quadratic inequalities is to skip the ‘sketch step’.
Sod’s Law states that even though you have a 50% chance of getting it right without a sketch (presuming you’ve factorised correctly), you will get it wrong.

7. Test Your Understanding
Solve 𝒙 𝟐 −𝒙−𝟔≤𝟎
𝑥+2 𝑥−3 ≤0 −2≤𝑥≤3 1 ?
Solve 𝟐 𝒙 𝟐 + 𝒙−𝟐 𝟐 >𝟑𝒙
2 𝑥 2 + 𝑥 2 −4𝑥+4>3𝑥 3 𝑥 2 −7𝑥+4>0 3𝑥−4 𝑥−1 >0 𝑥<1 𝑜𝑟 𝑥> 4 3 2 ?

8. Exercises ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 Solve the following:
𝑥 2 −5𝑥−6> 𝒙<−𝟏 𝒐𝒓 𝒙>𝟔 𝑥 2 +3𝑥−4≤ −𝟒≤𝒙≤𝟏 𝑥 2 −9≥ 𝒙≤−𝟑 𝒐𝒓 𝒙≥𝟑 𝑥 2 +8𝑥+12< −𝟔<𝒙<−𝟐 𝑥 2 −3𝑥−10≥ 𝒙≤−𝟐 𝒐𝒓 𝒙≥𝟓
𝑥+3 2 ≤ −𝟕≤𝒙≤𝟏
4 𝑥 2 ≥ 𝒙≤− 𝟑 𝟐 𝒐𝒓 𝒙≥ 𝟑 𝟐
𝑦 2 +𝑦> 𝒚<−𝟒 𝒐𝒓 𝒚>𝟑
16− 𝑥 2 < 𝒙<−𝟒 𝒐𝒓 𝒙>𝟒
𝑥 𝑥+2 2 < −𝟓<𝒙<𝟐
10− 2𝑥+1 2 ≥𝑥 − 𝟗 𝟒 ≤𝒙≤𝟏
(Requires knowledge of sketching cubics/quartics)
𝑥−1 𝑥−2 𝑥−3 > 𝟏<𝒙<𝟐 𝒐𝒓 𝒙>𝟑
𝑥−1 𝑥−2 𝑥−3 < 𝒙<𝟏 𝒐𝒓 𝟐<𝒙<𝟐
𝑥 2 −9 𝑥 2 −4 > 𝒙<−𝟑 𝒐𝒓 −𝟐<𝒙<𝟐 𝒐𝒓 𝒙>𝟑 a ? b ? 3
Solve the following:
2 𝑥 2 +3𝑥+1≤ −𝟏≤𝒙≤− 𝟏 𝟐
3 𝑥 2 −7𝑥−6> 𝒙<− 𝟐 𝟑 𝒐𝒓 𝒙>𝟑
4 𝑥 2 −4𝑥+1> 𝒙≠ 𝟏 𝟐 c ? ? a d ? ? b e ? c ? 2 a ? b ? c ? d ? e ? f ? N ? a b ? c ?