Bounds and Inequalities: Mathematics That Describes What Is Possible
An equation identifies exact matches. An inequality describes a region of possibility.
Additional Mathematics frequently asks more than “what is x?”. A quadratic may need to stay positive. A trigonometric expression has a maximum and minimum. A parameter may be restricted to intervals that produce real roots. A model may only be feasible for positive dimensions. A derivative sign can show where a function is increasing. R-form can produce a bound without solving for every angle.
This guide develops inequality and bound reasoning as a connected system. The learner moves among algebraic sign charts, completed squares, discriminants, trigonometric ranges, R-form, derivative signs and contextual feasibility.
AI Extraction Box: The Bound Loop
object → natural range/sign structure → inequality condition → solve intervals → test endpoints → interpret feasible region.
- Quadratic sign: use roots, opening direction or completed square.
- Quadratic always positive: a>0 and Δ<0.
- Quadratic non-negative: a>0 and Δ≤0.
- Trig bounds: −1≤sinθ,cosθ≤1.
- R-form: if expression=Rcos(θ−α), it lies between −R and R.
- Derivative sign: f′>0 increasing, f′<0 decreasing.
- Feasibility: combine mathematical inequality with contextual restrictions.
- Extremum: identify maximum/minimum and whether endpoints also matter.
Linear Inequalities: Direction Matters
The basic rule remains essential: multiplying or dividing by a negative number reverses the inequality sign.
−2x<6
x>−3.
Many later parameter inequalities fail because this simple reversal is forgotten inside more complicated algebra.
Quadratic Inequalities Through Sign Structure
Suppose:
(x−2)(x−5)>0.
The critical points 2 and 5 divide the number line into intervals. The product is positive when both factors have the same sign:
- x<2: both negative → positive;
- 2<x<5: opposite signs → negative;
- x>5: both positive → positive.
Therefore:
x<2 or x>5.
For ≥0, endpoints would be included.
Graph Interpretation of a Quadratic Inequality
Solving f(x)>0 asks where the graph y=f(x) lies above the x-axis. Solving f(x)<0 asks where it lies below. This graph view gives a strong cross-check for the number-line sign method.
For an upward-opening quadratic with roots 2 and 5, the graph is below the axis between roots and above outside them. The algebra and graph tell the same story.
Completed Square Gives Bounds Directly
Because a square is non-negative, completed-square form exposes a bound immediately.
Example:
x²−6x+11=(x−3)²+2.
Since (x−3)²≥0:
x²−6x+11≥2.
The minimum value 2 occurs at x=3. No calculus is needed.
Worked Example 1: Parameter for Positivity
Find k such that x²−4x+k is positive for all real x.
Because the leading coefficient is positive, require no real roots:
Δ<0
16−4k<0
k>4.
Alternatively:
x²−4x+k=(x−2)²+(k−4).
For strict positivity for all x, require k−4>0. The two representations agree.
Trigonometric Bounds
The unit circle gives:
−1≤sinθ≤1, −1≤cosθ≤1.
Therefore:
- 3+2sinθ lies between 1 and 5;
- 5−4cosθ lies between 1 and 9;
- sin²θ lies between 0 and 1.
These bounds can answer parameter or modelling questions without solving for θ.
Worked Example 2: Range of a Trig Expression
Find the range of y=7−3cosθ.
Since −1≤cosθ≤1:
−3≤−3cosθ≤3.
Add 7:
4≤y≤10.
R-Form Produces Sharp Trig Bounds
For a cosθ+b sinθ, write it as Rcos(θ−α) or equivalent. Since cosine lies between −1 and 1, the expression lies between −R and R.
Example:
3cosθ+4sinθ=5cos(θ−α).
Therefore:
−5≤3cosθ+4sinθ≤5.
The bound is exact and attainable.
Worked Example 3: Bound with a Vertical Shift
Find the maximum and minimum of 2+5cos(θ−α).
Because −1≤cos≤1:
−5≤5cos(θ−α)≤5.
Add 2:
−3≤2+5cos(θ−α)≤7.
Minimum −3, maximum 7.
Derivative Signs Create Interval Bounds on Behaviour
If f′(x)>0 on an interval, f is increasing there. If f′(x)<0, it is decreasing. Solving derivative inequalities partitions the domain into behavioural regions.
For f(x)=x³−3x:
f′(x)=3x²−3=3(x−1)(x+1).
Therefore f′>0 for x<−1 or x>1 and f′<0 for −1<x<1. The derivative inequality describes where the function rises and falls.
Extremal Reasoning with Calculus
To find a maximum/minimum in a differentiable model:
- state the valid domain;
- solve f′=0 for stationary candidates;
- classify using sign change or second derivative;
- compare endpoints if seeking a global extremum on a closed interval;
- interpret the feasible optimum.
A local stationary point does not automatically give the global maximum/minimum over a restricted interval.
Worked Example 4: Endpoint Comparison
Let f(x)=−x²+6x+2 on 0≤x≤5.
f′=−2x+6=0 gives x=3. Evaluate:
- f(0)=2;
- f(3)=11;
- f(5)=7.
Hence global maximum is 11 at x=3, and global minimum is 2 at x=0 over the stated interval.
The endpoints matter because the question restricts the domain.
Feasibility in Geometry and Modelling
A model may produce algebraically valid but geometrically impossible values. If x is a cut length from a rectangle, constraints may require 0<x<half the shorter side. If x is a radius, x>0. If t is elapsed time, t≥0.
Feasible-region reasoning combines inequalities from the mathematics and the context.
Inequalities in Exponential and Logarithmic Contexts
Because eˣ>0 for all real x, expressions containing a positive multiple of eˣ inherit useful sign information. Logarithmic functions require positive inputs and are increasing for base a>1, so inequality transformations may use monotonicity when appropriate.
For example, if ln x>ln5 and x>0, then x>5 because ln is increasing on its domain.
Bounds as Verification
Bounds can catch impossible answers quickly:
- a computed sine of 1.2 is impossible;
- a value below a completed-square minimum is impossible;
- a model quantity outside its physical range deserves checking;
- an exponential output with positive coefficient cannot be negative.
Knowing what cannot happen is a powerful checking tool.
Bounds Decision Tree
- Quadratic inequality? factor/sign chart or graph.
- Need quadratic bound? complete the square.
- Parameter controls root existence? discriminant inequality.
- Trig expression? basic range or R-form.
- Function behaviour? derivative inequality.
- Global extremum on interval? stationary candidates plus endpoints.
- Physical model? intersect mathematical and contextual constraints.
- Checking result? compare with known natural bounds.
Common Failure Modes
| Error | Cause | Repair |
|---|---|---|
| inequality sign not reversed after dividing negative | linear rule forgotten | mark reversal explicitly |
| endpoints omitted from ≥ inequality | strict/non-strict language blurred | test equality cases |
| quadratic positivity solved with roots only | graph opening/discriminant meaning not used | combine a sign with Δ condition |
| trig maximum exceeds natural amplitude | range ignored | use −1≤sin,cos≤1 or R-form |
| stationary point called global maximum automatically | domain/endpoints ignored | compare all feasible candidates |
| algebraic optimum outside physical range accepted | feasibility not checked | state contextual inequalities first |
A 50-Minute Bounds Session
- 10 minutes: quadratic sign charts and endpoint control.
- 8 minutes: completed-square lower/upper bounds.
- 8 minutes: parameter inequalities via discriminant.
- 8 minutes: trig ranges and R-form bounds.
- 8 minutes: derivative sign intervals and extrema.
- 8 minutes: feasibility constraints in models.
What Mastery Looks Like
- The learner treats inequalities as regions, not failed equations.
- The learner controls strict and non-strict endpoints.
- The learner uses completed square, discriminants and sign charts strategically.
- The learner derives trig bounds from natural ranges and R-form.
- The learner uses derivative signs to describe behavioural intervals.
- The learner compares endpoints for global extrema where required.
- The learner combines mathematical and physical constraints into feasible sets.
- The learner uses known bounds to reject impossible numerical answers.
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