Alright, let's go through both parts of the problem step-by-step.
### Problem 18: Simplification
We need to simplify the expression:
[tex]\[ \frac{1 - 5x}{1 - 5x} \][/tex]
1. Observe that the numerator and the denominator are the same:
[tex]\[ 1 - 5x \][/tex]
2. Since any non-zero number divided by itself equals 1, we can simplify this fraction directly:
[tex]\[
\frac{1 - 5x}{1 - 5x} = 1
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[ 1 \][/tex]
### Problem 20: Solving the Equation
We are given this equation to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{(x+1)(x+2)}{(x+11)(x-2)} = 1
\][/tex]
To solve this equation, let's follow these steps:
1. Cross-multiply to eliminate the fraction:
[tex]\[
(x + 1)(x + 2) = (x + 11)(x - 2)
\][/tex]
2. Expand both sides of the equation:
[tex]\[
x^2 + 3x + 2 = x^2 + 9x - 22
\][/tex]
3. Subtract [tex]\( x^2 \)[/tex] from both sides to eliminate the quadratic term:
[tex]\[
3x + 2 = 9x - 22
\][/tex]
4. Rearrange the equation to isolate [tex]\( x \)[/tex]:
[tex]\[
2 + 22 = 9x - 3x
\][/tex]
[tex]\[
24 = 6x
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{24}{6} = 4
\][/tex]
Thus, the solution to the equation is:
[tex]\[ x = 4 \][/tex]
### Summary
1. The simplified form of the expression [tex]\(\frac{1 - 5x}{1 - 5x}\)[/tex] is [tex]\( 1 \)[/tex].
2. The solution to the equation [tex]\(\frac{(x+1)(x+2)}{(x+11)(x-2)} = 1\)[/tex] is [tex]\( x = 4 \)[/tex].
