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≤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, find the bounds on −x.

Example 2

medium
If 2≤x≤6 and 1≤y≤3, find the bounds on the product xy (all positive).

Example 3

hard
If −2≤x≤3 and −4≤y≤1, find the tightest bounds on xy.

Example 4

medium
If 3≤x≤5, find the bounds on x2.

Example 5

challenge
For all real x, prove x2−4x+7≥3 and state the lower bound.

Example 6

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

Example 7

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

Example 8

medium
For f(x)=x2 on [−3,2], find the upper and lower bounds of f.

Example 9

medium
If 1≤x≤4 and 2≤y≤5 with both positive, find bounds on x/y.

Example 10

medium
If ∣x−4∣≤3, write the bounds on x.

Example 11

medium
A rounded measurement reads 7 cm to the nearest cm. What are the bounds on the true length L?

Example 12

hard
Show that for all real x, x2≥0. What is the greatest lower bound (infimum) of x2?

Example 13

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

Example 14

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

Example 15

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

Example 16

medium
Find bounds on 2x+3 given −1≤x≤4.

Example 17

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

Example 18

easy
Write 'x is at most 12 and at least 4' as a bounded inequality.

Example 19

medium
If −2≤x≤5 and 1≤y≤4, find the tightest bounds on x+y.

Example 20

medium
For f(x)=1x on [2,5], find the bounds of f.
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