MathematicsChapter 5: Algebraic Fractions

Rational Equations

State excluded values.

Learning objectives

Explain rational equations using the correct definitions, conditions, and notation.
Apply exclude, clear, solve, check correctly in routine and exam-style questions.
Show each important step, simplify the result, and check that it satisfies the original conditions.

Definition

Formula

x = -2

Where / symbols

RATIONAL EQUATIONS: State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original. This mixes Chapter 2 with this chapter.
CLEAR: 12/(x^2-16) - 24/(x-4) = 3. Bottoms (x-4)(x+4) and (x-4). LCD (x-4)(x+4). x != +/-4.
CHECK: A solution x=-2 that avoids +/-4 can stand. If algebra produced x=4, reject it. Checking in the cleared equation is not enough if you divided information away - check the original.

Overview

Four steps, in order:

Clear

12x²-16 - 24x-4 = 3. Bottoms (x-4)(x+4) and (x-4). LCD (x-4)(x+4). x != +/-4.

Examples x != 4, x != -4 · Times LCD · Get a polynomial

Check

A solution x=-2 that avoids +/-4 can stand. If algebra produced x=4, reject it. Checking in the cleared equation is not enough if you divided information away - check the original.

Examples Reject excluded · Keep good roots · Show the substitution

WHAT — RATIONAL EQUATIONS

State excluded values. Multiply every term by the LCD to clear fractions. Solve the polynomial (often linear or quadratic). Check each candidate in the original. This mixes Chapter 2 with this chapter. The symbols and conditions are part of the mathematics, so include them when stating the rule.

WHY — Why this topic matters

Rational Equations matters because rational expressions combine algebra with fraction rules. Factoring, restrictions, and checking are essential because an apparently simple answer may still exclude some values.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through CLEAR, then CHECK. Worked: 12x² - 16 - 24x - 4 = 3 [outline]. x^2 - 16 = (x-4)(x+4) excluded: x != 4 and x != -4 multiply through by (x-4)(x+4) 12 - 24(x+4) = 3(x-4)(x+4) solve the polynomial check: if a root is +/-4, reject it a valid root from the chapter list is x = -2 Finish by checking signs, restrictions, units, and whether the answer matches the question.

Worked example

12x² - 16 - 24x - 4 = 3 [outline].
x² - 16 = (x-4)(x+4)
excluded: x != 4 and x != -4
multiply through by (x-4)(x+4)
12 - 24(x+4) = 3(x-4)(x+4)
solve the polynomial
check: if a root is +/-4, reject it
a valid root from the chapter list is x = -2

In simple words

Never start by 'cancelling x-4' across an equation if that hides a possible extraneous root. Multiply by LCD, then check.

Key points

exclude, clear, solve, check
LCD of all bottoms
original bottoms

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.