Determining an Equation for a Given Graph Notes

\begin{definition}[Rational Expression] A rational expression is a fraction where both the numerator and denominator are polynomials. \end{definition}

\begin{note} The denominator of a rational expression cannot be zero. \end{note}

To add or subtract rational expressions, we need to:

1. Find a common denominator.
2. Rewrite each expression with the common denominator.
3. Add or subtract the numerators.
4. Simplify the resulting expression.

Adding Rational Expressions with the Same Denominator

If the rational expressions have the same denominator, we can add or subtract them by combining the numerators.

\begin{example} Example 1:

Simplify $\frac{3x}{x+2} + \frac{2}{x+2}$.

1. The denominators are the same: $x+2$.
2. Combine the numerators: $3x + 2$.
3. The result is $\frac{3x + 2}{x+2}$. \end{example}

Adding Rational Expressions with Different Denominators

If the rational expressions have different denominators, we need to:

1. Find the least common denominator (LCD).
2. Rewrite each expression with the LCD.
3. Combine the numerators.

\begin{example} Example 2:

Simplify $\frac{2}{x} + \frac{3}{x+1}$.

1. The denominators are different: $x$ and $x+1$.
2. The LCD is $x(x+1)$.
3. Rewrite each expression with the LCD:

• $\frac{2}{x} = \frac{2(x+1)}{x(x+1)} = \frac{2x + 2}{x(x+1)}$
• $\frac{3}{x+1} = \frac{3x}{x(x+1)}$

1. Combine the numerators: $\frac{2x + 2 + 3x}{x(x+1)} = \frac{5x + 2}{x(x+1)}$. \end{example}

\begin{warning} Be careful when distributing negative signs in subtraction. \end{warning}

\begin{example} Example 3:

Simplify $\frac{3x}{x+2} - \frac{x-2}{x+2}$.

1. The denominators are the same: $x+2$.
2. Subtract the numerators: $3x - (x - 2) = 3x - x + 2 = 2x + 2$.
3. The result is $\frac{2x + 2}{x+2}$. \end{example}

\begin{self_review}

1. Simplify $\frac{4}{x} + \frac{3}{x+2}$.
2. Simplify $\frac{5x}{x+1} - \frac{2}{x+1}$.
3. Simplify $\frac{2}{x+1} + \frac{3}{x+2}$. \end{self_review}