Radicals and Rational Expressions

Solving Rational Equations

Solve x/(x-2)+3=2/(x-2). State domain x≠2, multiply by (x-2) to clear denominators, solve 4x=8 → x=2. Since x=2 is excluded by the domain, the solution is extraneous.

solve-rational-equation Always note domain exclusions (denominator zeros) first. After clearing denominators, check every candidate against the ORIGINAL equation and domain — candidates excluded by the domain are extraneous.

Example

Clear denominators, solve, then reject candidates excluded by the domain.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

xx−2+3=2x−2\frac{x}{x- 2 }+ 3 =\frac{ 2 }{x- 2 }

Step 2 — State domain

State the domain first: x cannot be 2.

x≠2x\ne \hl{2}

Step 3 — Clear denominators

Multiply by the LCD to get 4x = 8.

(4)x=8( \hl{4} )x= \hl{8}

Step 4 — Solve candidate

Divide: 8 / 4 = 2.

x=2x= \hl{2}

Step 5 — Check candidate

Check the domain: x = 2 is not allowed because x cannot be 2.

x=2x≠2x= \hl{2} \quad x\ne \hl{2}

Step 6 — Verdict

Reject the candidate as extraneous.

extraneous\hl{extraneous}