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Evaluating Rational Expressions

Substitute values into fractions with variables, spot undefined values, and use real-world rational formulas.

Practice this skill 22 min Evaluate rational expressions Leer en español
Key idea

What you'll learn

To evaluate a rational expression means to replace the variable with a number and work out the value. For example, \dfrac{3x + 5}{x - 2} at x = 4 is \dfrac{17}{2} = 8.5.

In this lesson you will learn to:

  • Substitute safely, using parentheses, especially with negative numbers.
  • Treat the fraction bar as a grouping symbol: work out the whole top and the whole bottom before dividing.
  • Recognize when an expression is undefined (a zero denominator).
  • Simplify first when it makes the arithmetic easier.
  • Use real-world rational formulas: average cost, combined work time, average speed.
  • Work backwards: find the input that gives a certain value.

Why it matters on the GED. You will often be handed a formula with a fraction in it — the average cost of making n items, or the time two workers need together — and asked to plug in numbers. These questions are straightforward if you follow the order of operations, and easy to miss if you rush. The test's wrong answers come from exactly those rushed moves.

Real life. A print shop charges a setup fee plus a per-shirt price. The average cost per shirt, \dfrac{\text{setup} + \text{price} \cdot n}{n}, drops as you order more. Evaluating that expression tells you what each shirt really costs.

Words to know

  • Evaluate — find the value of an expression for given values of the variables.
  • Substitute — replace a variable with a number. Substituting x = -3 into x^2 gives (-3)^2 = 9.
  • Rational expression — a fraction with polynomials on the top and bottom, such as \dfrac{x^2 - 1}{2x + 3}.
  • Numerator / denominator — the top / bottom of the fraction.
  • Fraction bar — the line in a fraction. It acts like parentheses around the whole top and the whole bottom.
  • Undefined — has no value. A fraction with a denominator of 0 is undefined.
  • Excluded value — a value of the variable that makes a denominator 0.
  • Order of operations — parentheses, exponents, multiplication and division (left to right), addition and subtraction (left to right).
  • Average cost — total cost ÷ number of items.
  • Rate — an amount per unit of time, such as miles per hour or jobs per hour.

All Math words in the glossary

Substituting step by step

Method

  1. Rewrite the expression with parentheses where each variable was.
  2. Put the numbers inside the parentheses.
  3. Work out the entire numerator (using order of operations).
  4. Work out the entire denominator.
  5. Divide, and simplify the fraction or change it to a decimal if asked.

Example. Evaluate \dfrac{3x + 5}{x - 2} for x = 4.

\frac{3(4) + 5}{(4) - 2} = \frac{12 + 5}{2} = \frac{17}{2} = 8.5

Negative numbers need parentheses. Evaluate \dfrac{x^2 - 1}{2x + 3} for x = -3.

\frac{(-3)^2 - 1}{2(-3) + 3} = \frac{9 - 1}{-6 + 3} = \frac{8}{-3} = -\frac{8}{3}

Without parentheses, -3^2 means -(3^2) = -9. That one slip turns the numerator into -10.

ExpressionValue
(-3)^29
-3^2-9
(-2)^3-8
-2(-4)8

The fraction bar groups. In \dfrac{75 + 6n}{n} with n = 25, the whole top is divided by 25:

\frac{75 + 6(25)}{25} = \frac{75 + 150}{25} = \frac{225}{25} = 9

It is not 75 + \dfrac{6(25)}{25} = 81.

The fraction bar groups. Correct: the whole top 75 + 6(25) = 225 is divided by 25, giving 9. Wrong: dividing only 6(25) by 25 gives 75 + 6 = 81.75 + 6(25)25= 225 ÷ 25 = 9✓75 + 6(25) ÷ 25= 75 + 6 = 81✗the bar's grouping was lost: only 6(25) got divided

Two variables. Substitute each one in its own parentheses. For \dfrac{a^2 - b}{a - b} with a = -3, b = 5: \dfrac{(-3)^2 - 5}{(-3) - 5} = \dfrac{4}{-8} = -0.5.

Undefined values and simplifying first

When is it undefined? A rational expression is undefined when its denominator equals 0. To find those values, set the denominator equal to 0 and solve.

Number line for (x − 5) over (x squared − x − 12) = (x − 5) over (x − 4)(x + 3). Open red circles at x = −3 and x = 4 mark where the denominator is 0, so the expression is undefined there. A green dot at x = 5 shows a zero numerator, where the value is simply 0.x − 5x² − x − 12 = (x − 4)(x + 3)−5−4−3−2−10123456undefinedundefinedvalue 0
  • \dfrac{x + 2}{x - 7}: x - 7 = 0, so it is undefined at x = 7.
  • \dfrac{x - 5}{x^2 - x - 12}: (x - 4)(x + 3) = 0, so it is undefined at x = 4 and x = -3.

A zero numerator is fine. \dfrac{0}{32} = 0. Only a zero denominator makes it undefined. (And \dfrac{0}{0} is undefined too.)

Simplify first to save work. If the numbers are messy, factor and cancel before substituting. Evaluate \dfrac{x^2 - 4x}{x^2 - 16} at x = 6:

\frac{x(x - 4)}{(x - 4)(x + 4)} = \frac{x}{x + 4} \quad\Rightarrow\quad \frac{6}{10} = 0.6

Direct substitution agrees: \dfrac{36 - 24}{36 - 16} = \dfrac{12}{20} = 0.6.

Careful: simplifying does not change where the original is undefined. \dfrac{x^2 - 16}{x^2 + 4x} = \dfrac{(x - 4)(x + 4)}{x(x + 4)} simplifies to \dfrac{x - 4}{x}, but at x = -4 the original has a zero denominator. The answer there is "undefined," not \dfrac{-8}{-4} = 2.

Checking an answer with a decimal input. For x = 4.8, \dfrac{x^2 + 2x - 15}{x^2 - 25} = \dfrac{(x + 5)(x - 3)}{(x + 5)(x - 5)} = \dfrac{x - 3}{x - 5} = \dfrac{1.8}{-0.2} = -9. Simplifying turned a calculator chore into mental math.

Real-world rational formulas

These formulas appear in GED word problems. You will usually be given the formula; your job is to substitute correctly.

Average cost per item. If a job has a fixed cost F and a cost v per item, the average cost for n items is

\bar{C} = \frac{F + vn}{n}

Example: $1200 setup and $15 per item, n = 80: \dfrac{1200 + 15(80)}{80} = \dfrac{2400}{80} = \$30 per item.

Combined work time. If one person (or machine) takes a hours for a job and another takes b hours, together they take

t = \frac{ab}{a + b}

Example: a = 6, b = 3: t = \dfrac{18}{9} = 2 hours. Together is always faster than the faster worker alone.

Average speed for a round trip. Going at r_1 and returning at r_2 over the same distance:

\bar{r} = \frac{2 r_1 r_2}{r_1 + r_2}

Example: 40 mph and 60 mph give \dfrac{2 \cdot 40 \cdot 60}{100} = 48 mph — not 50.

Round-trip average speed, using 120 miles each way as an example: going at 40 mph takes 3 hours and returning at 60 mph takes 2 hours, so 240 miles in 5 hours is 48 mph, not 50. This matches 2 times 40 times 60 over 100 = 48.go: 120 mi at 40 mph = 3 hback: 60 mph = 2 h0 h1 h2 h3 h4 h5 h240 mi ÷ 5 h = 48 mph (not 50)2(40)(60)/(40 + 60) = 48: more time is spent at the slower speed

Working backwards. If you know the result and need the input, set the expression equal to the result and solve. Average cost \dfrac{1200 + 15n}{n} = 20: multiply both sides by n, so 1200 + 15n = 20n, giving 1200 = 5n and n = 240 items.

Formula

Formulas

\text{Average cost} = \frac{\text{fixed cost} + (\text{cost per item})(n)}{n} \text{Combined time} = \frac{ab}{a + b} \qquad \text{Average round-trip speed} = \frac{2r_1 r_2}{r_1 + r_2} d = rt \quad\Rightarrow\quad t = \frac{d}{r}
  • n = number of items; a, b = hours each worker needs alone; r_1, r_2 = the two speeds; d = distance, r = rate, t = time.

On the GED formula sheet: d = rt and "total cost = (number of units) × (price per unit)." The average-cost, combined-time and round-trip formulas are not on the sheet; when the GED uses them, it usually prints the formula in the question. Knowing them lets you check that an answer makes sense.

Worked example

Example 1 (easy): Basic substitution

Evaluate \dfrac{3x + 5}{x - 2} when x = 4.

Example 1 substitution template: the whole numerator 3(4) + 5 = 17 is boxed, the whole denominator (4) − 2 = 2 is boxed, then divide: 17/2 = 8.5.3(4) + 5 = 17(4) − 2 = 2172= 8.51. whole top 2. whole bottom3. divide
  1. Substitute: \dfrac{3(4) + 5}{(4) - 2}.
  2. Numerator: 12 + 5 = 17.
  3. Denominator: 4 - 2 = 2.
  4. Divide: \dfrac{17}{2} = 8.5.

Answer: \dfrac{17}{2} or 8.5

Check: 8.5 \times 2 = 17 ✓, and 17 is the numerator.

Worked example

Example 2: A negative input

Evaluate \dfrac{x^2 - 1}{2x + 3} when x = -3.

Example 2 with x = −3: the numerator (−3) squared − 1 = 8 and the denominator 2(−3) + 3 = −3, so the value is 8/(−3) = −8/3. Below: (−3) squared = 9 but −3 squared without parentheses = −9.(−3)² − 1 = 82(−3) + 3 = −38−3= −8/3(−3)² = 9−3² = −9with ( ): square −3no ( ): square 3 only
  1. Substitute with parentheses: \dfrac{(-3)^2 - 1}{2(-3) + 3}.
  2. Numerator: (-3)^2 = 9, so 9 - 1 = 8.
  3. Denominator: 2(-3) = -6, so -6 + 3 = -3.
  4. Divide: \dfrac{8}{-3} = -\dfrac{8}{3} \approx -2.67.

Answer: -\dfrac{8}{3}

Check: -\dfrac{8}{3} \times (-3) = 8 ✓

Worked example

Example 3: Simplify first

Evaluate \dfrac{x^2 - 4x}{x^2 - 16} when x = 6.

Example 3: x(x − 4) over (x − 4)(x + 4). The common factor (x − 4) is crossed out, leaving x/(x + 4). Substituting x = 6 gives 6/10 = 0.6.x(x − 4)(x − 4)(x + 4)=xx + 466 + 4= 0.6x = 6Factor, cancel (x − 4), then substitute into the smaller fraction
  1. Factor: top = x(x - 4), bottom = (x - 4)(x + 4).
  2. Cancel (x - 4) (allowed because x = 6 does not make it zero): \dfrac{x}{x + 4}.
  3. Substitute: \dfrac{6}{6 + 4} = \dfrac{6}{10}.

Answer: 0.6

Check: Direct substitution: \dfrac{36 - 24}{36 - 16} = \dfrac{12}{20} = 0.6 ✓

Worked example

Example 4 (real world): Two painters

Dana can paint a room in 6 hours. Eli can paint the same room in 3 hours. Working together, they need t = \dfrac{ab}{a + b} hours. How long will it take them together?

Example 4 bar model: the room is split into 6 equal sixths. Each hour Dana paints 1 sixth (blue) and Eli paints 2 sixths (green), so together they paint half the room per hour. After 2 hours the whole room is painted: t = ab/(a + b) = 18/9 = 2 hours.DanaEliEliDanaEliEliroomhour 1: 1/6 + 2/6 = 1/2hour 2: another 1/2Dana: 1/6 room per hour (6 h alone)Eli: 1/3 = 2/6 room per hour (3 h alone)t = ab/(a + b) = 18/9 = 2 hours
  1. a = 6, b = 3.
  2. Numerator: ab = 6 \cdot 3 = 18.
  3. Denominator: a + b = 6 + 3 = 9.
  4. t = \dfrac{18}{9} = 2.

Answer: 2 hours

Check: In 2 hours Dana paints \tfrac{2}{6} = \tfrac{1}{3} of the room and Eli paints \tfrac{2}{3}. Together \tfrac{1}{3} + \tfrac{2}{3} = 1 whole room ✓

Worked example

Example 5 (real world, work backwards): Average cost

A small company spends $1200 to set up a production run, plus $15 per lamp. The average cost per lamp is \dfrac{1200 + 15n}{n} dollars. (a) Find the average cost for 80 lamps. (b) How many lamps must be made to bring the average cost down to $20?

Graph of average cost (1200 + 15n)/n for n from 40 to 400 lamps. The curve falls as n grows and levels toward the dashed line at 15 dollars per lamp. Marked points: 80 lamps cost 30 dollars each (part a) and 240 lamps cost 20 dollars each (part b).01020304050080160240320400number of lamps, naverage cost ($ per lamp)$15 per lamp(80, $30)(240, $20)
  1. (a) \dfrac{1200 + 15(80)}{80} = \dfrac{1200 + 1200}{80} = \dfrac{2400}{80} = 30.
  2. (b) Set the expression equal to 20: \dfrac{1200 + 15n}{n} = 20.
  3. Multiply both sides by n: 1200 + 15n = 20n.
  4. Subtract 15n: 1200 = 5n, so n = 240.

Answer: (a) $30 per lamp; (b) 240 lamps

Check: \dfrac{1200 + 15(240)}{240} = \dfrac{1200 + 3600}{240} = \dfrac{4800}{240} = 20 ✓

Calculator tip

Calculator tip: let the fraction template group for you

On the TI-30XS MultiView, the n/d key creates a stacked fraction. Everything you type in the top box stays together, and everything in the bottom box stays together, so the calculator respects the fraction bar automatically.

  1. Press n/d, type the whole numerator, press the down arrow, type the whole denominator, then press the right arrow to leave the fraction.
  2. For negative inputs use the (−) key and parentheses: ( (−) 3 ) x² gives 9.
  3. Press toggle to switch the answer between a fraction and a decimal.

If you use the ÷ key instead, you must add parentheses: ( 75 + 6 × 25 ) ÷ 25 → 9. Without them the calculator computes 75 + 6 and gives 81.

When not to use it: simple values like \dfrac{6}{10} are faster in your head, and finding where an expression is undefined is algebra, not arithmetic.

Common mistake

Common mistakes

  • Ignoring the fraction bar. Wrong: \dfrac{75 + 6(25)}{25} = 75 + 6 = 81. Right: \dfrac{225}{25} = 9. Finish the whole top and the whole bottom before dividing.
  • Squaring a negative without parentheses. Wrong: for x = -2, x^2 = -2^2 = -4. Right: x^2 = (-2)^2 = 4.
  • Calling a zero numerator "undefined." Wrong: \dfrac{0}{32} is undefined. Right: \dfrac{0}{32} = 0. Only a zero denominator is undefined.
  • Using the simplified form at an excluded value. Wrong: \dfrac{x^2 - 16}{x^2 + 4x} at x = -4 equals 2. Right: it is undefined there.
  • Averaging rates. Wrong: going 40 mph and returning 60 mph averages 50 mph. Right: \dfrac{2(40)(60)}{100} = 48 mph, because you spend more time at the slower speed.
  • Inverting the fraction. Wrong: \dfrac{x + 6}{x - 1} at x = 3 gives \dfrac{2}{9}. Right: \dfrac{9}{2}. Keep top on top.
Test strategy

GED test strategy

  • Write the parentheses first, then fill in the numbers. It takes five seconds and prevents sign errors.
  • Estimate. Combined work time must be less than either single time; an average cost must be more than the per-item cost. Eliminate choices that break these rules.
  • Simplify first when the numbers are large or decimal; substitute directly when they are small.
  • "Undefined" questions: set only the denominator to zero; ignore the numerator.
  • Work-backwards questions: plug each answer choice into the formula. The one that produces the given value is correct.
  • Read the units in the question (hours or minutes, dollars per item or total dollars).

Check yourself

Evaluate rational expressionsNo calculator

What is the value of \dfrac{x + 6}{x - 1} when x = 3?

Answer choices
Hint

Work out the entire numerator and the entire denominator first, then divide.

Check yourself

Evaluate rational expressionsNo calculatorFill in

Evaluate \dfrac{2x^2 - 5}{x + 4} when x = -2.

Enter your answer as a decimal or fraction.

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Put -2 in parentheses. Remember that (-2)^2 is positive.

Summary

  • Substitute into parentheses, then evaluate the whole numerator and the whole denominator before dividing.
  • (-3)^2 = 9 but -3^2 = -9: parentheses matter.
  • An expression is undefined where its original denominator is 0; a zero numerator just gives 0.
  • Simplify first when it makes the numbers easier.
  • Know the real-world forms: average cost \dfrac{F + vn}{n}, combined time \dfrac{ab}{a + b}, round-trip speed \dfrac{2r_1r_2}{r_1 + r_2}.
  • To work backwards, set the expression equal to the target, multiply by the denominator, and solve.
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