Non-Permissible Values & Simplifying Rational Expressions

Simplify rational expressions and find restrictions. Math 20-1 Alberta.

Lesson 6.2 of Rational Expressions and Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Finding Non-Permissible Values. Learn what non-permissible values are, why they exist, and how to find them by factoring the denominator and setting each factor equal to zero.
• Simplifying Rational Expressions. Factor both numerator and denominator completely, then cancel common factors — including additive inverses — to reduce rational expressions to lowest terms.
• Working Backwards and Applications. Build rational expressions from given NPVs, test whether two expressions are equivalent, and use rational expressions to find unknown dimensions in geometry problems.

Inside the lesson — a free preview

What is a Non-Permissible Value?

Definition: Non-Permissible Value (NPV). A non-permissible value (NPV) is any value of the variable that would make the denominator equal to zero. Since dividing by zero is undefined, those values are excluded from the domain of the expression.

How to Find NPVs — 5 Steps.

Critical Rule. Always find NPVs from the original denominator, before any simplifying. Even if a factor cancels out later, it still creates an NPV that must be stated.

Two Key Properties

Property 1 — Anything Divided by Itself Equals 1. Any non-zero polynomial divided by itself equals 1.

Example: (x + 5)/(x + 5) = 1, where x ≠ -5

Property 2 — Anything Divided by Its Additive Inverse Equals -1. The additive inverse of an expression is the same expression with every term negated.

Example: The additive inverse of (x - 3) is -(x - 3) = -x + 3 = 3 - x.

So (x - 3)/(3 - x) = -1, where x ≠ 3.

Pattern to watch for: If you see (a - b) over (b - a), they are additive inverses — the whole fraction simplifies to -1.

How to Simplify — 3 Steps.

Worked example: a–b — Simple Denominators

Part a) — (1)/(5x^2). Step 1 — Identify the denominator: 5x^2.

Step 2 — Set the denominator equal to zero.

5x^2 = 0

Step 3 — Solve for x. Divide both sides by 5:

x^2 = 0

Take the square root of both sides:

x = 0

Answer: x ≠ 0

Part b) — (1)/(2x - 3). Step 1 — Identify the denominator: 2x - 3.

Step 2 — Set the denominator equal to zero.

2x - 3 = 0

Step 3 — Solve for x. Add 3 to both sides:

2x = 3

Divide both sides by 2:

x = (3)/(2)

Answer: x ≠ (3)/(2)

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