Question

write the following expression (\frac{x + 5}{x - 6} + \frac{x + 2}{x - 1} - 1) in the standard form of a rational expression:

Explanation:

Step1: Find a common denominator

The denominators are \(x - 6\), \(x - 1\), and \(1\). The common denominator is \((x - 6)(x - 1)\).
Rewrite each term with the common denominator:
\(\frac{(x + 5)(x - 1)}{(x - 6)(x - 1)}+\frac{(x + 2)(x - 6)}{(x - 6)(x - 1)}-\frac{(x - 6)(x - 1)}{(x - 6)(x - 1)}\)

Step2: Expand the numerators

Expand each numerator:

• \((x + 5)(x - 1)=x^2 - x + 5x - 5=x^2 + 4x - 5\)
• \((x + 2)(x - 6)=x^2 - 6x + 2x - 12=x^2 - 4x - 12\)
• \((x - 6)(x - 1)=x^2 - x - 6x + 6=x^2 - 7x + 6\)

So the expression becomes:
\(\frac{x^2 + 4x - 5 + x^2 - 4x - 12-(x^2 - 7x + 6)}{(x - 6)(x - 1)}\)

Step3: Simplify the numerator

Combine like terms in the numerator:
\[

$$\begin{align*} &x^2 + 4x - 5 + x^2 - 4x - 12 - x^2 + 7x - 6\\ =&(x^2 + x^2 - x^2)+(4x - 4x + 7x)+(- 5 - 12 - 6)\\ =&x^2 + 7x - 23 \end{align*}$$

\]

Step4: Write the final rational expression

The simplified expression is \(\frac{x^2 + 7x - 23}{(x - 6)(x - 1)}\) (we can also expand the denominator as \(x^2 - 7x + 6\) if needed, but the form with factored denominator is also a standard rational expression form).

Answer:

\(\frac{x^2 + 7x - 23}{(x - 6)(x - 1)}\) (or \(\frac{x^2 + 7x - 23}{x^2 - 7x + 6}\))