Constraints (Meta) Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Solve 1xโˆ’2=3\frac{1}{x-2} = 3 and identify all constraints on xx before solving.

Solution

  1. 1
    Constraint: xโˆ’2โ‰ 0x - 2 \ne 0, i.e., xโ‰ 2x \ne 2 (denominator cannot be zero).
  2. 2
    Multiply both sides by (xโˆ’2)(x-2): 1=3(xโˆ’2)1 = 3(x-2).
  3. 3
    Expand: 1=3xโˆ’61 = 3x - 6, so 3x=73x = 7, giving x=73x = \frac{7}{3}.
  4. 4
    Check constraint: 73โ‰ 2\frac{7}{3} \ne 2. Valid.

Answer

x=73x = \frac{7}{3}
Constraints limit the set of allowable values. For rational expressions, the denominator must be non-zero. Checking the solution against constraints is always required.

About Constraints (Meta)

Constraints are conditions, rules, or boundaries that restrict which values or solutions are allowed in a mathematical problem, narrowing an infinite space of possibilities to a manageable set.

Learn more about Constraints (Meta) โ†’

More Constraints (Meta) Examples

Example 2 medium

An integer [formula] satisfies two constraints: [formula] and [formula], and also [formula] is prime

Example 3 easy

State the domain constraint for [formula] and solve for [formula] when [formula].

Example 4 medium

A rectangle has area 36 cmยฒ and integer side lengths. List all constraints and find all valid dimens

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