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When two rational expressions have denominators that are **opposites** of each other — such as $$(a - b)$$ and $$(b - a)$$ — they can be rewritten to share a common denominator by using the algebraic identity:

$$b - a = -(a - b)$$

This identity shows that reversing the order of a subtraction is equivalent to multiplying by $$-1$$. To apply this technique, multiply the numerator and denominator of the fraction whose denominator is in the "opposite" form by $$-1$$. This transforms the denominator into the same expression as the other fraction's denominator, creating a common denominator so that the standard addition or subtraction rule can be used.

For instance, to add $$\frac{m}{m - 3} + \frac{5}{3 - m}$$, recognize that $$(3 - m)$$ and $$(m - 3)$$ are opposites. Multiply the numerator and denominator of the second fraction by $$-1$$: $$\frac{5}{3 - m} = \frac{5 \cdot (-1)}{(3 - m)\cdot(-1)} = \frac{-5}{m - 3}$$. Now both fractions share the denominator $$(m - 3)$$, and they can be combined: $$\frac{m + (-5)}{m - 3} = \frac{m - 5}{m - 3}$$.

Adding and Subtracting Rational Expressions Whose Denominators Are Opposites

Add two rational expressions whose denominators are opposites of each other:

$$\frac{4u - 1}{3u - 1} + \frac{u}{1 - 3u}$$

**Step 1 — Recognize the opposite denominators.** The denominators $$3u - 1$$ and $$1 - 3u$$ are opposites because $$1 - 3u = -(3u - 1)$$. To create a common denominator, multiply the numerator and denominator of the second fraction by $$-1$$:

$$\frac{4u - 1}{3u - 1} + \frac{u \cdot (-1)}{(1 - 3u) \cdot (-1)}$$

**Step 2 — Simplify the second fraction.** Multiplying gives $$\frac{-u}{3u - 1}$$. Now both fractions share the denominator $$3u - 1$$:

$$\frac{4u - 1}{3u - 1} + \frac{-u}{3u - 1}$$

**Step 3 — Add the numerators over the common denominator:**

$$\frac{4u - 1 + (-u)}{3u - 1} = \frac{4u - 1 - u}{3u - 1}$$

**Step 4 — Combine like terms in the numerator.** Group the $$u$$-terms: $$4u - u = 3u$$:

$$\frac{3u - 1}{3u - 1}$$

**Step 5 — Simplify.** The numerator and denominator are identical, so the expression equals $$1$$.

This example illustrates a case where adding rational expressions with opposite denominators produces a numerator that matches the denominator exactly, causing the entire expression to simplify to $$1$$.

Adding $$\frac{4u-1}{3u-1} + \frac{u}{1-3u}$$

Subtract two rational expressions whose denominators are opposites, then factor and simplify:

$$\frac{m^2 - 6m}{m^2 - 1} - \frac{3m + 2}{1 - m^2}$$

**Step 1 — Recognize the opposite denominators.** The denominators $$m^2 - 1$$ and $$1 - m^2$$ are opposites because $$1 - m^2 = -(m^2 - 1)$$. Multiply the numerator and denominator of the second fraction by $$-1$$:

$$\frac{m^2 - 6m}{m^2 - 1} - \frac{(3m + 2)(-1)}{(1 - m^2)(-1)}$$

**Step 2 — Simplify the second fraction.** Multiplying gives $$\frac{-3m - 2}{m^2 - 1}$$. Both fractions now share the denominator $$m^2 - 1$$:

$$\frac{m^2 - 6m}{m^2 - 1} - \frac{-3m - 2}{m^2 - 1}$$

**Step 3 — Subtract the numerators over the common denominator:**

$$\frac{m^2 - 6m - (-3m - 2)}{m^2 - 1}$$

**Step 4 — Distribute the negative sign.** Subtracting $$(-3m - 2)$$ reverses the signs: $$-(-3m) = +3m$$ and $$-(-2) = +2$$:

$$\frac{m^2 - 6m + 3m + 2}{m^2 - 1}$$

**Step 5 — Combine like terms.** Group the $$m$$-terms: $$-6m + 3m = -3m$$:

$$\frac{m^2 - 3m + 2}{m^2 - 1}$$

**Step 6 — Factor the numerator and denominator.** The numerator $$m^2 - 3m + 2$$ factors as $$(m - 1)(m - 2)$$, since $$(-1) + (-2) = -3$$ and $$(-1)(-2) = 2$$. The denominator $$m^2 - 1$$ is a difference of squares: $$(m - 1)(m + 1)$$:

$$\frac{(m - 1)(m - 2)}{(m - 1)(m + 1)}$$

**Step 7 — Simplify by removing the common factor.** Cancel the shared factor $$(m - 1)$$:

$$\frac{m - 2}{m + 1}$$

This example combines the opposite-denominators technique with polynomial subtraction, trinomial factoring, and the difference of squares pattern. After converting the denominators to match, the subtraction and sign distribution produce a trinomial numerator that factors and shares a common binomial factor with the denominator, enabling further simplification.

Subtracting $$\frac{m^2-6m}{m^2-1} - \frac{3m+2}{1-m^2}$$

Subtract two rational expressions where one denominator is a factorable trinomial and the other is opposite to one of its factors, then find the LCD, combine, and simplify: 

$$\frac{-3n - 9}{n^2 + n - 6} - \frac{n + 3}{2 - n}$$

Step 1 — Factor the first denominator. The trinomial $$n^2 + n - 6$$ factors as $$(n - 2)(n + 3)$$, since $$(-2) + 3 = 1$$ and $$(-2)(3) = -6$$:

$$\frac{-3n - 9}{(n - 2)(n + 3)} - \frac{n + 3}{2 - n}$$

Step 2 — Handle the opposite denominators. The expressions $$(n - 2)$$ and $$(2 - n)$$ are opposites because $$2 - n = -(n - 2)$$. Multiply the numerator and denominator of the second fraction by $$-1$$:

$$\frac{-3n - 9}{(n - 2)(n + 3)} - \frac{(-1)(n + 3)}{(-1)(2 - n)} = \frac{-3n - 9}{(n - 2)(n + 3)} + \frac{n + 3}{n - 2}$$

The subtraction of a fraction with the opposite denominator converts to addition once the denominator is corrected.

Step 3 — Find the LCD and rewrite. The denominators are now $$(n - 2)(n + 3)$$ and $$(n - 2)$$. The LCD is $$(n - 2)(n + 3)$$. The first fraction already has the LCD. Multiply the second fraction's numerator and denominator by $$(n + 3)$$:

$$\frac{-3n - 9}{(n - 2)(n + 3)} + \frac{(n + 3)(n + 3)}{(n - 2)(n + 3)}$$

Expand the second numerator: $$(n + 3)(n + 3) = n^2 + 6n + 9$$.

Step 4 — Add the numerators over the common denominator:

$$\frac{-3n - 9 + n^2 + 6n + 9}{(n - 2)(n + 3)} = \frac{n^2 + 3n}{(n - 2)(n + 3)}$$

Step 5 — Factor the numerator and simplify. Extract the GCF from the numerator: $$n^2 + 3n = n(n + 3)$$. Cancel the shared factor $$(n + 3)$$:

$$\frac{n(n + 3)}{(n - 2)(n + 3)} = \frac{n}{n - 2}$$

This example combines the opposite-denominators technique with the standard unlike-denominators procedure. After converting the opposite denominator using $$\frac{-1}{-1}$$, the two fractions still do not share a common denominator — one additional step of rewriting with the LCD is required before the numerators can be combined. The problem also illustrates that factoring the first denominator may reveal a factor that is the opposite of the second denominator, a pattern that is not always immediately obvious.

Subtracting $$\frac{-3n-9}{n^2+n-6} - \frac{n+3}{2-n}$$

A financial analyst is comparing two growth models. Model A uses the denominator $(r - 0.05)$ and Model B uses the denominator $(0.05 - r)$. State the specific algebraic identity that allows the analyst to rewrite the denominator $(0.05 - r)$ in a form that reveals it is the 'opposite' of $(r - 0.05)$.

Identifying the Identity for Opposite Denominators

A benefits administrator is reconciling two different pension contribution models. Model A uses a denominator of (E - D), where E represents total earnings and D represents mandatory deductions. Model B uses a denominator of (D - E). To combine these models, the administrator needs to rewrite the denominator of Model B to match Model A.

State the specific algebraic identity that justifies this transformation. Additionally, describe the two-step mathematical procedure required to rewrite a fraction with the denominator (D - E) so that it shares the common denominator (E - D) without changing the value of the expression.

Standardizing Opposite Denominators in Benefits Administration

Based on the scenario, what specific algebraic identity must the planner recall to rewrite the denominator $$(35 - v)$$ as an expression that shares the common denominator $$(v - 35)$$?

Traffic Flow Data Integration

To add two rational expressions whose denominators are opposites, we can multiply one of the fractions by $$\frac{-1}{-1}$$ to create a common denominator. For example, consider the expression $$\frac{7}{d} + \frac{5}{-d}$$. Since $$d$$ and $$-d$$ are opposites, we multiply the second fraction by $$\frac{-1}{-1}$$:

$$\frac{5}{-d} \cdot \frac{-1}{-1} = \frac{-5}{d}$$

Now the denominators are the same, and we can add the numerators:

$$\frac{7}{d} + \frac{-5}{d} = \frac{7 + (-5)}{d} = \frac{2}{d}$$

This simple example illustrates the general principle that when denominators are opposites, multiplying one fraction by $$\frac{-1}{-1}$$ yields a common denominator, allowing for addition or subtraction.

Adding $$\frac{7}{d} + \frac{5}{-d}$$

Subtract two rational expressions where one denominator is the opposite of a binomial factor of the other:
$$\frac{3x - 1}{x^2 - 5x - 6} - \frac{2}{6 - x}$$

Step 1 — Factor the denominator and resolve the opposites. Factor the first denominator: $$x^2 - 5x - 6 = (x - 6)(x + 1)$$. The second denominator is $$6 - x$$. Because $$x - 6$$ and $$6 - x$$ are opposites, multiply the second fraction by $$\frac{-1}{-1}$$:
$$\frac{2}{6 - x} \cdot \frac{-1}{-1} = \frac{-2}{x - 6}$$
Rewrite the subtraction:
$$\frac{3x - 1}{(x - 6)(x + 1)} - \frac{-2}{x - 6}$$

Step 2 — Simplify the signs. Subtracting a negative is the same as adding a positive:
$$\frac{3x - 1}{(x - 6)(x + 1)} + \frac{2}{x - 6}$$

Step 3 — Find the LCD and rewrite. The LCD is $$(x - 6)(x + 1)$$. The first fraction already has the LCD. Multiply the numerator and denominator of the second fraction by $$(x + 1)$$:
$$\frac{3x - 1}{(x - 6)(x + 1)} + \frac{2(x + 1)}{(x - 6)(x + 1)}$$
Distribute the $$2$$ in the second numerator: $$2(x + 1) = 2x + 2$$.
$$\frac{3x - 1}{(x - 6)(x + 1)} + \frac{2x + 2}{(x - 6)(x + 1)}$$

Step 4 — Add the numerators. Combine the numerators over the common denominator:
$$\frac{3x - 1 + 2x + 2}{(x - 6)(x + 1)}$$
Combine like terms to get the simplified result:
$$\frac{5x + 1}{(x - 6)(x + 1)}$$

Subtracting $$\frac{3x-1}{x^2-5x-6} - \frac{2}{6-x}$$

Subtract two rational expressions involving opposite binomial denominators and simplify:
$$\frac{-2y - 2}{y^2 + 2y - 8} - \frac{y - 1}{2 - y}$$

Step 1 — Factor the denominator and resolve the opposites. Factor the first denominator: $$y^2 + 2y - 8 = (y + 4)(y - 2)$$. The second denominator, $$2 - y$$, is the opposite of $$y - 2$$. Multiply the second fraction by $$\frac{-1}{-1}$$:
$$\frac{y - 1}{2 - y} \cdot \frac{-1}{-1} = \frac{-(y - 1)}{y - 2}$$
Rewrite the subtraction:
$$\frac{-2y - 2}{(y + 4)(y - 2)} - \frac{-(y - 1)}{y - 2}$$

Step 2 — Simplify the signs. Subtracting a negative is the same as adding a positive ($$a - (-b) = a + b$$):
$$\frac{-2y - 2}{(y + 4)(y - 2)} + \frac{y - 1}{y - 2}$$

Step 3 — Find the LCD and rewrite. The LCD is $$(y + 4)(y - 2)$$. Multiply the numerator and denominator of the second fraction by $$(y + 4)$$:
$$\frac{-2y - 2}{(y + 4)(y - 2)} + \frac{(y - 1)(y + 4)}{(y + 4)(y - 2)}$$
Expand the second numerator using FOIL: $$(y - 1)(y + 4) = y^2 + 3y - 4$$.
$$\frac{-2y - 2}{(y + 4)(y - 2)} + \frac{y^2 + 3y - 4}{(y + 4)(y - 2)}$$

Step 4 — Add the numerators. Combine over the common denominator:
$$\frac{-2y - 2 + y^2 + 3y - 4}{(y + 4)(y - 2)}$$
Combine like terms and rearrange into standard form:
$$\frac{y^2 + y - 6}{(y + 4)(y - 2)}$$

Step 5 — Factor and simplify. Factor the new numerator: $$y^2 + y - 6 = (y + 3)(y - 2)$$.
$$\frac{(y + 3)(y - 2)}{(y + 4)(y - 2)}$$
Cancel the common factor $$(y - 2)$$ to find the final simplified expression:
$$\frac{y + 3}{y + 4}$$

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