Bounds Formula

The Formula

a \leq x \leq b

When to use: Temperature tomorrow will be between 60F and 75F. Those are bounds.

Quick Example

2 \leq x \leq 7 Here 2 is the lower bound and 7 is the upper bound.

Notation

a \leq x \leq b means x is between a and b inclusive; (a, b) or [a, b] in interval notation

What This Formula Means

The upper and lower limits within which a quantity must lie; often expressed as a \leq x \leq b.

Temperature tomorrow will be between 60F and 75F. Those are bounds.

Formal View

L \leq x \leq U \iff x \in [L, U], \; \text{where } L = \inf(S) \text{ and } U = \sup(S)

Worked Examples

Example 1

medium
For \(f(x) = -x^2 + 6x - 5\), find the maximum value (upper bound) on \([0, 5]\).

Solution

  1. 1
    Find the vertex: \(x = -b/(2a) = -6/(2 \times -1) = 3\).
  2. 2
    \(f(3) = -9 + 18 - 5 = 4\).
  3. 3
    Check endpoints: \(f(0) = -5\), \(f(5) = -25+30-5=0\).
  4. 4
    Maximum is \(f(3) = 4\).

Answer

Maximum value = 4 at \(x = 3\)
For a downward parabola, the vertex gives the maximum. The upper bound of \(f\) on \([0,5]\) is 4.

Example 2

hard
Show that for all real \(x\), \(x^2 \geq 0\). What is the greatest lower bound (infimum) of \(x^2\)?

Common Mistakes

  • Using a strict bound (<) when the endpoint should be included (\leq), or vice versa
  • Confusing upper and lower bounds — the lower bound is the smallest allowed value, not the largest
  • Forgetting that a bound only limits from one direction — x \leq 7 allows any value from -\infty up to 7

Why This Formula Matters

Essential for estimation, measurement error analysis, and finding maximum/minimum values in optimization.

Frequently Asked Questions

What is the Bounds formula?

The upper and lower limits within which a quantity must lie; often expressed as a \leq x \leq b.

How do you use the Bounds formula?

Temperature tomorrow will be between 60F and 75F. Those are bounds.

What do the symbols mean in the Bounds formula?

a \leq x \leq b means x is between a and b inclusive; (a, b) or [a, b] in interval notation

Why is the Bounds formula important in Math?

Essential for estimation, measurement error analysis, and finding maximum/minimum values in optimization.

What do students get wrong about Bounds?

Bounds can be strict (< and >) or inclusive (\leq and \geq).

What should I learn before the Bounds formula?

Before studying the Bounds formula, you should understand: inequality intuition.

← Back to Bounds See All Examples Practice Problems