7.3 Rational Expressions: Simplifying, Evaluating & Writing

Why This Section Matters

Rational expressions are the smallest indicator cluster in Assessment Target A.1 — only 3 of A.1's 10 indicators (A.1.h, A.1.i, A.1.j) — but they test a distinct and frequently missed skill: expressions in which the variable appears in the denominator. Real-world rate problems such as miles per gallon, cost per person, and speed (distance ÷ time) are naturally modeled as rational expressions, so this indicator connects directly to word-problem items across both the Quantitative and Algebraic content areas of the GED. Because a rational expression is undefined whenever its denominator equals zero, this section also introduces a reasoning skill — identifying restricted values — that has no equivalent in Sections 7.1 or 7.2.

Core Vocabulary

• A rational expression is a ratio (fraction) of two polynomials, such as (x + 3)/(x − 5).
• A restricted value (also called an excluded value) is any value of the variable that makes the denominator equal zero. Because division by zero is undefined, restricted values must always be excluded from the domain of the expression.
• Simplifying a rational expression means factoring the numerator and denominator completely, then canceling only shared factors — never individual terms.

Simplifying, Multiplying, and Dividing Rational Expressions (A.1.h, part 1)

To simplify, factor completely and cancel common factors:

(x² − 25)/(x + 5) = [(x + 5)(x − 5)] / (x + 5) = x − 5, for x ≠ −5

The most damaging trap on this indicator is canceling terms that are not full factors of the entire numerator or denominator. For example, (x + 3)/x is not equal to 3 — you cannot "cancel the x's," because x is only one term of the numerator, not a factor of the whole numerator. Cancellation is only valid when the numerator and denominator share a factor that multiplies the entire expression, as in the example above where (x + 5) is a genuine factor of both.

Multiplying rational expressions: multiply numerators together and denominators together, then simplify.

(2x/3) × (9/x²) = 18x / 3x² = 6/x, for x ≠ 0

Dividing rational expressions: multiply by the reciprocal of the second expression (keep–change–flip), then simplify.

(x/4) ÷ (x²/8) = (x/4) × (8/x²) = 8x / 4x² = 2/x, for x ≠ 0

Adding and Subtracting Rational Expressions (A.1.h, part 2)

Rational expressions add and subtract exactly like numeric fractions: with like denominators, combine the numerators directly.

3/x + 2/x = 5/x

With unlike denominators, find a common denominator first, rewrite each fraction over it, and then combine numerators:

3/x + 2/(x + 1) = [3(x + 1) + 2x] / [x(x + 1)] = (5x + 3) / [x(x + 1)]

Skipping the common-denominator step and simply adding numerators and denominators separately (a common shortcut error that produces 5/(2x + 1)) is mathematically invalid and a frequent wrong-answer choice on GED items testing this skill.

Evaluating Rational Expressions (A.1.i)

Substitute the given integer for the variable and compute, exactly as in Section 7.1 — but now also check whether the substituted value is a restricted value.

Evaluate (x + 4)/(x − 2) at x = 6: (6 + 4)/(6 − 2) = 10/4 = 2.5

Evaluate the same expression at x = 2: the denominator becomes 2 − 2 = 0, so the expression is undefined at x = 2. GED items sometimes ask this directly, in the form "for what value of x is this expression undefined?" To answer, set the denominator equal to 0 and solve — never the numerator. For (2x + 1)/(x − 7), the expression is undefined when x − 7 = 0, so x = 7 is the restricted value, regardless of what the numerator equals at that point.

Writing Rational Expressions from Context (A.1.j)

Real-world rates translate directly into rational expressions using the pattern rate = quantity ÷ measure:

• "Miles per gallon when a car travels m miles using (g − 2) gallons of gas" → m/(g − 2)
• "Cost per ticket when a $240 venue fee is split evenly among (n + 4) group members" → 240/(n + 4)

In the ticket example, n = −4 is a restricted value — it is not a realistic group size in context, and it also makes the denominator zero, confirming algebraically that it must be excluded. Whenever a GED item asks you to write a rational expression from a scenario, identify the total or fixed quantity for the numerator and the varying group, distance, or count for the denominator before assembling the fraction.

Exam Scenario

A GED item states that a charter bus company splits a flat $240 rental fee evenly among (n + 4) passengers, where n is the number of passengers beyond the first 4 who signed up early. It asks for the cost per passenger when n = 8. Substituting n = 8 into the denominator first gives n + 4 = 12, so the cost per passenger is 240 ÷ 12 = $20. Expect a distractor built from applying order of operations incorrectly, such as 240 ÷ 8 + 4 = 34 (dividing before adding the 4), or from ignoring the +4 term entirely, giving 240 ÷ 8 = 30.