Grade by Grade
A. Characteristics of Functions · Functions and Equivalent Expressions

Adding and Subtracting Rational Expressions

A3.3About 75 minutes
Learning goals
  • I can find the lowest common denominator of rational expressions by factoring.
  • I can add and subtract rational expressions and simplify the result.
  • I can state the restrictions on the variable.
  • I can write and simplify a rational expression for a real situation.

Words to know

common denominator
A denominator that every fraction in a sum can be rewritten with.
lowest common denominator (LCD)
The simplest common denominator: every different factor, each to the highest power it has in any one denominator.
equivalent fractions
Fractions with the same value, such as 1/2 and 3/6; made by multiplying top and bottom by the same non-zero factor.

You can only add like pieces

You cannot add a half of a pizza and a third of a pizza by adding the tops and bottoms: 1/2 + 1/3 is not 2/5. You first cut both into sixths, a common denominator, and then count: 3/6 + 2/6 = 5/6.

Adding and subtracting rational expressions works the same way, and the hard part is the same too: finding the lowest common denominator (LCD).

1/21/33/62/6sum= 5/6halves and thirdsdon't line upsixths: a commondenominator
Halves and thirds do not line up, but both can be cut into sixths. Then 3 sixths + 2 sixths = 5 sixths.

Steps

  1. Factor every denominator.
  2. State the restrictions.
  3. Find the LCD: the product of every different factor, each raised to the highest power it has in any one denominator.
  4. Rewrite each expression with the LCD by multiplying its numerator and denominator by the missing factors.
  5. Add or subtract the numerators. Put brackets around each numerator, especially after a minus sign.
  6. Simplify the numerator, then factor it and divide out any common factor with the denominator.
Worked example

Example 1: Monomial denominators

Simplify 3/(4x) + 5/(6x²).

  1. Restriction: x ≠ 0.
  2. LCD of 4x and 6x²: numbers 12 (lowest common multiple of 4 and 6), variable x² (highest power). LCD = 12x².
  3. 3/(4x) = (3 × 3x)/(4x × 3x) = 9x/(12x²). 5/(6x²) = (5 × 2)/(6x² × 2) = 10/(12x²).
  4. Add: (9x + 10)/(12x²), x ≠ 0.

(9x + 10)/(12x²), x ≠ 0

Worked example

Example 2: Binomial denominators

Simplify 2/(x + 3) − 1/(x − 2).

  1. Restrictions: x ≠ −3, 2. The denominators share no factor, so LCD = (x + 3)(x − 2).
  2. Rewrite: 2(x − 2)/[(x + 3)(x − 2)] − 1(x + 3)/[(x + 3)(x − 2)].
  3. Subtract numerators with brackets: [2(x − 2) − (x + 3)] = 2x − 4 − x − 3 = x − 7.
  4. Result: (x − 7)/[(x + 3)(x − 2)], x ≠ −3, 2.
  5. Check with x = 0: 2/3 − 1/(−2) = 2/3 + 1/2 = 7/6; answer −7/(3 × −2) = 7/6 ✓.

(x − 7)/[(x + 3)(x − 2)]

The bracket rule

When you subtract a numerator, the minus sign applies to every term in it: −(x + 3) = −x − 3. Leaving out the brackets is the most common mistake in this lesson.

Worked example

Example 3: Factor first

Simplify x/(x² − 4) + 3/(x + 2).

  1. Factor: x² − 4 = (x − 2)(x + 2). Restrictions: x ≠ ±2.
  2. LCD = (x − 2)(x + 2). The second fraction needs the factor (x − 2): 3(x − 2)/[(x − 2)(x + 2)].
  3. Add: [x + 3(x − 2)]/[(x − 2)(x + 2)] = (4x − 6)/[(x − 2)(x + 2)].
  4. Factor the numerator to check for cancelling: 2(2x − 3). No common factor. Result: 2(2x − 3)/[(x − 2)(x + 2)].

2(2x − 3)/[(x − 2)(x + 2)], x ≠ ±2

x² − x − 6 = (x − 3)(x + 2)x² − 4x + 3 = (x − 3)(x − 1)LCD = (x − 3)(x + 2)(x − 1)every different factor,shared factors only once(x − 3) is shared, so it appears only once in the LCD.
Building the LCD for Example 4. A factor that appears in both denominators is used only once.
Worked example

Example 4: Trinomials, and a simplification at the end

Simplify (x + 1)/(x² − x − 6) − (x − 1)/(x² − 4x + 3).

  1. Factor: (x + 1)/[(x − 3)(x + 2)] − (x − 1)/[(x − 3)(x − 1)].
  2. Restrictions: x ≠ 3, −2, 1. LCD = (x − 3)(x + 2)(x − 1).
  3. Numerators: (x + 1)(x − 1) − (x − 1)(x + 2) = (x² − 1) − (x² + x − 2) = x² − 1 − x² − x + 2 = −x + 1.
  4. −x + 1 = −(x − 1), so the result is −(x − 1)/[(x − 3)(x + 2)(x − 1)].
  5. Divide out (x − 1): −1/[(x − 3)(x + 2)], x ≠ 3, −2, 1.

−1/[(x − 3)(x + 2)], x ≠ −2, 1, 3

Worked example

Example 5: Opposite denominators

Simplify 5/(x − 4) + 3/(4 − x).

  1. 4 − x = −(x − 4), so 3/(4 − x) = −3/(x − 4).
  2. Now 5/(x − 4) − 3/(x − 4) = 2/(x − 4), x ≠ 4.

2/(x − 4), x ≠ 4

Worked example

Example 6: A canoe trip

A canoeist paddles 12 km across a lake at x km/h, then 12 km back at (x + 1) km/h because the wind has calmed. Write and simplify an expression for the total time.

  1. time = distance ÷ speed, so total time = 12/x + 12/(x + 1).
  2. LCD = x(x + 1): [12(x + 1) + 12x]/[x(x + 1)] = (24x + 12)/[x(x + 1)] = 12(2x + 1)/[x(x + 1)] hours.
  3. In context x > 0. Check x = 3: 12/3 + 12/4 = 4 + 3 = 7 h; the expression gives 12(7)/(3 × 4) = 84/12 = 7 ✓.

12(2x + 1)/[x(x + 1)] hours

Just for fun

Adding tops and bottoms is not totally useless. For 1/2 and 1/3 it gives 2/5, which mathematicians call the mediant, and it always lands between the two fractions (1/3 < 2/5 < 1/2). It is just never their sum. Great at sitting in the middle, terrible at adding.

Hands-on activity

Recipe fractions

You need: measuring cups (1/2, 1/3 and 1/4 cups if you have them), a bowl, a 1-cup measure and dry rice or water.

  1. Pour 1/2 cup and then 1/3 cup of rice into the bowl. Pour the bowl into the 1-cup measure. Is it more or less than 1 cup?
  2. Predict the amount with fractions: 1/2 + 1/3 = ? Compare with 2/5.
  3. Repeat with 1/3 + 1/4.
  4. Now write and simplify 1/x + 1/(x + 1). Substitute x = 2 and x = 3 and match them with steps 1 and 3.

What to look for: 1/2 + 1/3 = 5/6 cup (a little under 1 cup, far more than 2/5). 1/3 + 1/4 = 7/12 cup. 1/x + 1/(x + 1) = (2x + 1)/[x(x + 1)]: x = 2 gives 5/6 and x = 3 gives 7/12.

Your turn

1

Simplify 2/(3a) + 5/(6a).

2

Simplify 4/x − 3/x².

3

Simplify 3/(x − 1) + 2/(x + 4).

4

Simplify 5/(x + 2) − 2/(x − 3).

5

Simplify 1/(x² − 9) + 2/(x + 3).

6

Simplify 7/(x − 5) − 2/(5 − x).

More practice: warm-up

1

Compute 1/4 + 1/6.

2

What is the LCD of 2x and 3x²?

3

Simplify 3/y + 4/y.

4

What is the LCD of (x + 1) and (x − 1)?

5

Simplify x/(x + 2) + 2/(x + 2).

More practice: on your own

1

Simplify (x + 2)/(x² − 1) − 1/(x − 1).

2

Simplify 3/(x² + 5x + 6) + 2/(x² + 4x + 4).

3

Simplify 2x/(x − 4) − 8/(x − 4).

4

Simplify 1/a + 1/b.

5

A hiker walks 6 km uphill at x km/h and 6 km downhill at 2x km/h. Write and simplify the total time.

Challenge

1

Simplify 2/(x² − x) − 3/(x² − 1) + 1/(x² + x).

2

Show that 1/n − 1/(n + 1) = 1/[n(n + 1)], and use it to find 1/(1 × 2) + 1/(2 × 3) + 1/(3 × 4) + 1/(4 × 5) quickly.

3

Find A and B so that A/(x − 1) + B/(x + 2) = (5x + 1)/[(x − 1)(x + 2)].

Check your understanding

Quiz

1.

What is the LCD of 3/(x + 4) and 5/(x² − 16)?

2.

Simplify 1/x + 1/(2x).

3.

Simplify 4/(x − 1) − 2/(x + 1).

4.

Which mistake gives 1/2 + 1/3 = 2/5?

5.

Simplify 3/(x − 2) + 3/(2 − x).

6.

When subtracting (x + 1)/(x − 3) − (2x − 5)/(x − 3), the numerator is:

For parents and teachers

Big idea: Expectation A3.3 includes adding and subtracting rational expressions and stating restrictions. It is the same process as adding fractions with unlike denominators, so a quick review of numeric fractions helps a lot.

How to help: Before starting, have your teen add 1/4 + 1/6 and explain each step. Then ask them to match each step to the algebra version. Encourage a numeric check (substitute x = 0 or x = 3) at the end.

Common difficulties:

  • Adding tops and bottoms separately.
  • Using a product of all denominators when a smaller LCD exists (correct but messy; the result then needs more simplifying).
  • Dropping the brackets when subtracting a numerator.
  • Forgetting to factor the final numerator to check for a common factor.