Practice Bounds in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

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

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

Showing a random 20 of 50 problems.

Example 1

medium
If −2≤x≤3-2\le x\le 3, find the bounds on −x-x.

Example 2

medium
If 2≤x≤62\le x\le 6 and 1≤y≤31\le y\le 3, find the bounds on the product xyxy (all positive).

Example 3

hard
If −2≤x≤3-2\le x\le 3 and −4≤y≤1-4\le y\le 1, find the tightest bounds on xyxy.

Example 4

medium
If 3≤x≤53\le x\le 5, find the bounds on x2x^2.

Example 5

challenge
For all real xx, prove x2−4x+7≥3x^2-4x+7\ge 3 and state the lower bound.

Example 6

hard
For f(x)=x3−3xf(x)=x^3-3x on [−2,2][-2,2], find the upper and lower bounds.

Example 7

hard
If 2<x<62<x<6 and 3<y<53<y<5, find the tightest open bounds on x+yx+y.

Example 8

medium
For f(x)=x2f(x)=x^2 on [−3,2][-3,2], find the upper and lower bounds of ff.

Example 9

medium
If 1≤x≤41\le x\le 4 and 2≤y≤52\le y\le 5 with both positive, find bounds on x/yx/y.

Example 10

medium
If ∣x−4∣≤3|x-4|\le 3, write the bounds on xx.

Example 11

medium
A rounded measurement reads 77 cm to the nearest cm. What are the bounds on the true length LL?

Example 12

hard
Show that for all real xx, x2≥0x^2 \geq 0. What is the greatest lower bound (infimum) of x2x^2?

Example 13

easy
Write the interval [2,9)[2,9) as an inequality on xx.

Example 14

challenge
By AM-GM, find the minimum of x+4xx+\frac{4}{x} for x>0x>0 and state the lower bound.

Example 15

hard
Greatest lower bound of the set {1/n:n∈Z+}\{1/n : n \in \mathbb{Z}^+\}?

Example 16

medium
Find bounds on 2x+32x+3 given −1≤x≤4-1\le x\le 4.

Example 17

easy
Give an integer that satisfies both x>4x>4 and x≤7x\le 7.

Example 18

easy
Write 'x is at most 1212 and at least 44' as a bounded inequality.

Example 19

medium
If −2≤x≤5-2\le x\le 5 and 1≤y≤41\le y\le 4, find the tightest bounds on x+yx+y.

Example 20

medium
For f(x)=1xf(x)=\frac{1}{x} on [2,5][2,5], find the bounds of ff.
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