Adding and Subtracting Algebraic Fractions

Adding and subtracting algebraic fractions follows the same fundamental rule as adding and subtracting numerical fractions:

The fractions must have a common denominator.

For example:

2/7 + 3/7 = 5/7

Because the denominators are already the same, we add the numerators.

But:

1/3 + 1/4

cannot be added directly. We first find a common denominator:

1/3 = 4/12

1/4 = 3/12

Then:

4/12 + 3/12 = 7/12

Algebraic fractions work in exactly the same way.

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The Basic Rule

If two fractions already have the same denominator:

a/c + b/c = (a + b)/c

Similarly:

a/c − b/c = (a − b)/c

The denominator stays the same.

Only the numerators are added or subtracted.

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Adding Fractions with the Same Denominator

Consider:

x/5 + 3/5

The denominator is already 5.

Combine the numerators:

(x + 3)/5

Therefore:

x/5 + 3/5 = (x + 3)/5

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Subtracting Fractions with the Same Denominator

Consider:

x/7 − 2/7

Combine the numerators:

(x − 2)/7

Therefore:

x/7 − 2/7 = (x − 2)/7

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Algebraic Denominators

Consider:

3/(x + 2) + 5/(x + 2)

The denominators are identical.

Add the numerators:

(3 + 5)/(x + 2)

Therefore:

8/(x + 2)

Restriction:

x ≠ −2

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Variables in the Numerators

Consider:

2x/(x − 4) + 3x/(x − 4)

The denominators are identical.

Add:

(2x + 3x)/(x − 4)

Simplify:

5x/(x − 4)

Restriction:

x ≠ 4

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Subtraction Requires Care

Consider:

5x/(x + 3) − 2x/(x + 3)

Combine:

(5x − 2x)/(x + 3)

Therefore:

3x/(x + 3)

Restriction:

x ≠ −3

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When the Denominators Are Different

Consider:

1/x + 1/3

We cannot simply add:

2/(x + 3)

That is incorrect.

Instead, find a denominator that both x and 3 divide into.

The common denominator is:

3x

Rewrite:

1/x = 3/(3x)

and:

1/3 = x/(3x)

Now add:

3/(3x) + x/(3x)

Therefore:

(x + 3)/(3x)

Restriction:

x ≠ 0

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Common Denominators

A common denominator is an expression that is a multiple of every denominator in the problem.

For:

1/4 + 1/6

possible common denominators include:

12, 24, 36, ...

The smallest is:

12

This is called the least common denominator, or LCD.

Using the LCD usually makes the calculation easier.

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Least Common Denominator

The least common denominator is the smallest expression containing every factor required by the denominators.

For example:

1/(2x) + 1/(3x)

The denominators are:

2x

and:

3x

The LCD is:

6x

because both 2x and 3x divide into 6x.

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Rewriting Equivalent Fractions

Suppose we have:

1/x

and want denominator 5x.

Multiply numerator and denominator by 5:

1/x × 5/5 = 5/(5x)

Therefore:

1/x = 5/(5x)

Multiplying numerator and denominator by the same nonzero expression does not change the value of the fraction.

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Why Equivalent Fractions Work

Consider:

2/3

Multiply numerator and denominator by 4:

2/3 × 4/4

Since:

4/4 = 1

we have not changed the fraction's value.

Therefore:

2/3 = 8/12

The same idea works with algebraic expressions.

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Adding Different Numerical Denominators

Consider:

x/3 + x/4

The LCD is:

12

Rewrite:

x/3 = 4x/12

x/4 = 3x/12

Now add:

4x/12 + 3x/12

= 7x/12

Therefore:

x/3 + x/4 = 7x/12

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Subtracting Different Numerical Denominators

Consider:

x/2 − x/5

LCD:

10

Rewrite:

x/2 = 5x/10

x/5 = 2x/10

Subtract:

5x/10 − 2x/10

Therefore:

3x/10

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Different Variable Denominators

Consider:

1/x + 1/y

The denominators are x and y.

The common denominator is:

xy

Rewrite:

1/x = y/(xy)

and:

1/y = x/(xy)

Add:

y/(xy) + x/(xy)

Therefore:

(x + y)/(xy)

Restrictions:

x ≠ 0

y ≠ 0

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Subtracting Different Variable Denominators

Consider:

1/x − 1/y

LCD:

xy

Rewrite:

1/x = y/(xy)

1/y = x/(xy)

Subtract:

y/(xy) − x/(xy)

Therefore:

(y − x)/(xy)

with:

x ≠ 0, y ≠ 0

Notice the order:

y − x

not:

x − y

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Numerical and Variable Factors

Consider:

1/(2x) + 1/(3x)

The LCD is:

6x

Rewrite:

1/(2x) = 3/(6x)

1/(3x) = 2/(6x)

Add:

3/(6x) + 2/(6x)

Therefore:

5/(6x)

Restriction:

x ≠ 0

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Another Example

Add:

3/(4x) + 5/(6x)

The LCD of 4x and 6x is:

12x

Rewrite:

3/(4x) = 9/(12x)

5/(6x) = 10/(12x)

Add:

19/(12x)

Therefore:

3/(4x) + 5/(6x) = 19/(12x)

with:

x ≠ 0

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Denominators Containing Different Variables

Consider:

2/(3x) + 5/(4y)

The LCD is:

12xy

Rewrite the first fraction:

2/(3x) × 4y/4y = 8y/(12xy)

Rewrite the second:

5/(4y) × 3x/3x = 15x/(12xy)

Add:

(8y + 15x)/(12xy)

Restrictions:

x ≠ 0

y ≠ 0

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Factor the Denominators First

When denominators contain algebraic expressions, factor them before finding the LCD.

Consider:

1/(x² − 4) + 1/(x + 2)

Factor:

x² − 4 = (x − 2)(x + 2)

So the denominators are:

(x − 2)(x + 2)

and:

x + 2

The LCD is:

(x − 2)(x + 2)

Factoring makes the relationship between the denominators visible.

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Complete Example with Factoring

Add:

1/(x² − 4) + 1/(x + 2)

Restrictions from the original denominators:

x ≠ 2

x ≠ −2

Factor:

1/[(x − 2)(x + 2)] + 1/(x + 2)

The first fraction already has the LCD.

Rewrite the second:

1/(x + 2) = (x − 2)/[(x − 2)(x + 2)]

Now add:

[1 + (x − 2)]/[(x − 2)(x + 2)]

Simplify the numerator:

x − 1

Therefore:

(x − 1)/[(x − 2)(x + 2)]

with:

x ≠ ±2

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Denominators That Share a Factor

Consider:

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

The denominators are:

x(x + 3)

and:

x + 3

The LCD is:

x(x + 3)

Rewrite:

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

Now add:

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

Therefore:

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

or:

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

Restrictions:

x ≠ 0, −3

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The LCD Does Not Repeat Shared Factors

Suppose the denominators are:

x(x + 2)

and:

3(x + 2)

We do not need:

3x(x + 2)²

The factor x + 2 is already shared.

The LCD is:

3x(x + 2)

Each necessary factor appears only as many times as required.

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Repeated Factors

Consider denominators:

(x + 1)

and:

(x + 1)²

The LCD is:

(x + 1)²

We need the highest required power of each factor.

Similarly, for:

x²

and:

x³

the LCD contains:

x³

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Example with Repeated Factors

Add:

1/(x + 1) + 2/(x + 1)²

LCD:

(x + 1)²

Rewrite the first fraction:

1/(x + 1) = (x + 1)/(x + 1)²

Now add:

(x + 1)/(x + 1)² + 2/(x + 1)²

Combine:

(x + 3)/(x + 1)²

Restriction:

x ≠ −1

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Finding the LCD Systematically

For complicated denominators:

1. Factor every denominator completely.
2. List each different factor.
3. Use each factor at its highest required power.
4. Include the least common multiple of numerical coefficients.

For example:

6x²(x + 1)

and:

4x(x + 1)²

Numerical LCD of 6 and 4:

12

Highest power of x:

x²

Highest power of x + 1:

(x + 1)²

Therefore the LCD is:

12x²(x + 1)²

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Subtracting Algebraic Fractions

Subtraction requires special care because the minus sign applies to the entire numerator of the second fraction.

Consider:

(x + 5)/7 − (x − 2)/7

Combine:

[(x + 5) − (x − 2)]/7

Now distribute the negative:

(x + 5 − x + 2)/7

Simplify:

7/7

Therefore:

1

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Why Parentheses Matter in Subtraction

Consider:

(2x + 3)/5 − (x − 4)/5

Correct:

[(2x + 3) − (x − 4)]/5

Distribute the negative:

(2x + 3 − x + 4)/5

Simplify:

(x + 7)/5

A common mistake would be to write:

2x + 3 − x − 4

This changes the sign incorrectly.

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Different Denominators with Subtraction

Simplify:

2/x − 3/(x + 1)

Restrictions:

x ≠ 0, −1

LCD:

x(x + 1)

Rewrite:

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

and:

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

Subtract:

[2(x + 1) − 3x]/[x(x + 1)]

Expand:

(2x + 2 − 3x)/[x(x + 1)]

Simplify:

(2 − x)/[x(x + 1)]

Therefore:

(2 − x)/[x(x + 1)]

with:

x ≠ 0, −1

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Addition Followed by Simplification

Sometimes the resulting numerator and denominator contain a common factor.

Consider:

1/x + 1/(x + 2)

LCD:

x(x + 2)

Rewrite:

(x + 2)/[x(x + 2)] + x/[x(x + 2)]

Add:

(2x + 2)/[x(x + 2)]

Factor numerator:

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

No common factors remain.

Therefore:

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

Restrictions:

x ≠ 0, −2

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Always Check Whether the Final Answer Simplifies

Consider:

x/(x + 2) + 2/(x + 2)

Add:

(x + 2)/(x + 2)

Simplify:

1

But the original denominator gives:

x ≠ −2

Therefore:

1, where x ≠ −2

Even though the denominator disappeared, the restriction remains.

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A Result That Cancels

Consider:

x/(x − 3) − 3/(x − 3)

Combine:

(x − 3)/(x − 3)

Simplify:

1

Restriction:

x ≠ 3

Therefore:

1, where x ≠ 3

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Do Not Cancel Before Combining Terms

Consider:

x/(x + 3) + 3/(x + 3)

You cannot cancel x with x + 3.

Instead combine the numerators:

(x + 3)/(x + 3)

Then simplify:

1

with:

x ≠ −3

Sometimes cancellation becomes possible after addition or subtraction.

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Adding Fractions with Factored Denominators

Simplify:

2/(x − 1) + 3/(x + 2)

Restrictions:

x ≠ 1, −2

LCD:

(x − 1)(x + 2)

Rewrite:

2(x + 2)/[(x − 1)(x + 2)]

and:

3(x − 1)/[(x − 1)(x + 2)]

Add:

[2(x + 2) + 3(x − 1)]/[(x − 1)(x + 2)]

Expand:

2x + 4 + 3x − 3

Simplify:

(5x + 1)/[(x − 1)(x + 2)]

No common factors remain.

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Subtracting with Factored Denominators

Simplify:

4/(x + 3) − 1/(x − 2)

Restrictions:

x ≠ −3, 2

LCD:

(x + 3)(x − 2)

Rewrite:

4(x − 2)/[(x + 3)(x − 2)]

and:

(x + 3)/[(x + 3)(x − 2)]

Subtract:

[4(x − 2) − (x + 3)]/[(x + 3)(x − 2)]

Expand:

4x − 8 − x − 3

Simplify:

3x − 11

Therefore:

(3x − 11)/[(x + 3)(x − 2)]

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Factoring Quadratic Denominators

Consider:

1/(x² + 5x + 6) + 1/(x + 2)

Factor:

x² + 5x + 6 = (x + 2)(x + 3)

So:

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

LCD:

(x + 2)(x + 3)

Rewrite:

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

Add:

(x + 4)/[(x + 2)(x + 3)]

Restrictions:

x ≠ −2, −3

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Denominators That Look Different but Are Related

Consider:

1/(x − 3) + 1/(3 − x)

Notice:

3 − x = −(x − 3)

Therefore:

1/(3 − x) = −1/(x − 3)

So:

1/(x − 3) − 1/(x − 3)

Therefore:

0

Restriction:

x ≠ 3

Recognizing reversed factors can make some problems much easier.

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Adding Three Algebraic Fractions

Consider:

1/x + 1/(2x) + 1/3

LCD:

6x

Rewrite:

1/x = 6/(6x)

1/(2x) = 3/(6x)

1/3 = 2x/(6x)

Add:

(6 + 3 + 2x)/(6x)

Therefore:

(2x + 9)/(6x)

Restriction:

x ≠ 0

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Multi-Step Example

Simplify:

2/(x + 1) + 1/(x − 1) − 3/(x² − 1)

Factor:

x² − 1 = (x − 1)(x + 1)

Restrictions:

x ≠ −1, 1

LCD:

(x − 1)(x + 1)

Rewrite:

2(x − 1)/[(x − 1)(x + 1)]

+ (x + 1)/[(x − 1)(x + 1)]

− 3/[(x − 1)(x + 1)]

Combine numerators:

2(x − 1) + (x + 1) − 3

Expand:

2x − 2 + x + 1 − 3

Simplify:

3x − 4

Therefore:

(3x − 4)/[(x − 1)(x + 1)]

with:

x ≠ ±1

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A Problem That Simplifies Completely

Simplify:

1/(x − 2) − 4/(x² − 4)

Factor:

x² − 4 = (x − 2)(x + 2)

Restrictions:

x ≠ 2, −2

LCD:

(x − 2)(x + 2)

Rewrite:

(x + 2)/[(x − 2)(x + 2)] − 4/[(x − 2)(x + 2)]

Combine:

(x + 2 − 4)/[(x − 2)(x + 2)]

Simplify:

(x − 2)/[(x − 2)(x + 2)]

Cancel:

1/(x + 2)

But retain the original restrictions:

x ≠ 2, −2

Final answer:

1/(x + 2), where x ≠ ±2

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Verifying by Substitution

Suppose:

1/x + 1/(x + 2)

simplifies to:

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

Choose an allowed value:

x = 2

Original:

1/2 + 1/4 = 3/4

Simplified:

2(3)/[2(4)]

= 6/8

= 3/4

The values agree.

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Verification Can Reveal Errors

Suppose someone claims:

1/x + 1/2 = 2/(x + 2)

Choose:

x = 2

Original:

1/2 + 1/2 = 1

Claimed result:

2/4 = 1/2

Since:

1 ≠ 1/2

the proposed simplification is incorrect.

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Algebraic Fractions in Science

Adding and subtracting algebraic fractions can appear when mathematical models or scientific formulas are combined.

For example, suppose two rates are:

r₁ = A/t₁

and:

r₂ = B/t₂

Their combined expression might involve:

A/t₁ + B/t₂

The common denominator is:

t₁t₂

So:

A/t₁ = At₂/(t₁t₂)

and:

B/t₂ = Bt₁/(t₁t₂)

Therefore:

A/t₁ + B/t₂ = (At₂ + Bt₁)/(t₁t₂)

This shows why common-denominator skills become useful in more advanced science and mathematics.

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Parallel Resistance

A well-known physics example is the relationship for two resistors in parallel:

1/R = 1/R₁ + 1/R₂

Find a common denominator:

R₁R₂

Then:

1/R = R₂/(R₁R₂) + R₁/(R₁R₂)

Therefore:

1/R = (R₁ + R₂)/(R₁R₂)

Taking reciprocals gives:

R = R₁R₂/(R₁ + R₂)

This is a practical application of adding algebraic fractions.

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Recognizing When the LCD Is Already Present

Consider:

2/[x(x + 4)] + 3/(x + 4)

The first denominator already contains:

x

and:

x + 4

Therefore the LCD is simply:

x(x + 4)

Do not unnecessarily multiply both denominators together and create:

x(x + 4)²

That denominator would work, but it is not the least common denominator and makes the algebra more difficult.

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Efficient Strategy

For algebraic fractions, use:

Restrict → Factor → LCD → Rewrite → Combine → Simplify → Verify

Restrict

Identify values that make original denominators zero.

Factor

Factor all denominators completely.

LCD

Identify the least common denominator.

Rewrite

Create equivalent fractions with the LCD.

Combine

Add or subtract the numerators.

Simplify

Expand or factor the numerator as necessary and cancel any common factors.

Verify

Check using substitution or algebraic reasoning.

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Common Mistakes

Mistake: Adding the denominators.

Incorrect:

a/b + c/d = (a + c)/(b + d)

This is not the rule for adding fractions.

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Mistake: Subtracting the denominators.

The denominator should be made common before subtraction.

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Mistake: Adding fractions before finding a common denominator.

For:

1/x + 1/3

you cannot simply write:

2/(x + 3)

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Mistake: Changing the denominator without changing the numerator.

If:

1/x

is rewritten with denominator 3x, multiply both numerator and denominator by 3:

3/(3x)

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Mistake: Forgetting parentheses when subtracting.

For:

(2x + 3)/5 − (x − 4)/5

write:

[(2x + 3) − (x − 4)]/5

before simplifying.

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Mistake: Cancelling terms before combining.

You may cancel factors, not terms separated by addition or subtraction.

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Mistake: Failing to factor denominators before finding the LCD.

Factoring often reveals shared factors and produces a much simpler LCD.

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Mistake: Multiplying entire denominators unnecessarily.

The product of denominators is a common denominator, but it may not be the least common denominator.

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Mistake: Forgetting restrictions.

Restrictions come from the original denominators and remain even if a factor later cancels.

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Mistake: Stopping before simplifying the final result.

After combining fractions, factor the numerator and denominator if possible and check for common factors.

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Key Terms

Algebraic fraction — fraction containing variables or algebraic expressions.

Common denominator — denominator shared by two or more fractions.

Least common denominator (LCD) — smallest expression that is a multiple of every denominator.

Equivalent fractions — fractions representing the same value despite having different numerators and denominators.

Numerator — expression above the fraction line.

Denominator — expression below the fraction line.

Factor — quantity multiplied by another quantity.

Common factor — factor shared by two or more expressions.

Factorization — rewriting an expression as a product of factors.

Restriction — variable value excluded because it makes an original denominator zero.

Common multiple — quantity that is a multiple of two or more expressions.

Simplification — rewriting an expression in an equivalent but simpler form.

Difference of squares — factoring pattern a² − b² = (a − b)(a + b).

Equivalent denominator — denominator created by multiplying the numerator and denominator by the same nonzero expression.

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Key Takeaways

• Algebraic fractions can only be added or subtracted directly when they have the same denominator.
• When denominators already match, keep the denominator and combine the numerators.
• Different denominators must first be rewritten using a common denominator.
• The least common denominator usually makes calculations easiest.
• Equivalent fractions are created by multiplying numerator and denominator by the same nonzero expression.
• Factor algebraic denominators before finding the LCD.
• Shared factors should appear only as many times as required in the LCD.
• For repeated factors, use the highest required power.
• Restrictions should be identified from the original denominators.
• Restrictions remain even when factors later cancel.
• Addition combines the complete numerators.
• Subtraction requires parentheses around the second numerator.
• A negative sign must be distributed to every term in a subtracted numerator.
• Never add or subtract denominators.
• Never cancel individual terms across addition or subtraction.
• Simplification usually occurs after the numerators have been combined.
• The final numerator may need to be expanded, combined, or factored.
• If the final numerator and denominator share a factor, it may then be cancelled.
• Some final expressions cannot be simplified further.
• Substitution provides a useful way to verify an answer.
• Common-denominator techniques are used in algebra, rational equations, rates, electrical circuits, science formulas, and many other mathematical models.

Check Your Understanding

1. Add x/7 + 3/7.
2. Subtract 5x/9 − 2x/9.
3. Add 3/(x + 4) + 2/(x + 4) and state the restriction.
4. Find the least common denominator of 1/x and 1/5.
5. Rewrite 1/x using denominator 5x.
6. Simplify x/3 + x/5.
7. Simplify x/4 − x/6.
8. Simplify 1/x + 1/y and state all restrictions.
9. Simplify 1/x − 1/y and explain why the order of the numerator matters.
10. Simplify 1/(2x) + 1/(3x).
11. Simplify 3/(4x) − 1/(6x).
12. Find the LCD of x(x + 2) and 3(x + 2).
13. Simplify 2/(x − 1) + 3/(x + 2).
14. Simplify 4/(x + 3) − 1/(x − 2).
15. Simplify 1/(x² − 9) + 1/(x + 3) and state all restrictions.
16. Simplify 1/(x + 1) + 2/(x + 1)².
17. Simplify x/(x − 4) − 4/(x − 4) and state the restriction.
18. Explain why 1/x + 1/3 ≠ 2/(x + 3).
19. Describe how substitution can be used to verify the addition or subtraction of algebraic fractions.
20. Simplify completely and state all restrictions:

2/(x + 2) + 1/(x − 2) − 8/(x² − 4).