Let's solve the equation [tex]\(\frac{1}{x+1} + \frac{1}{x+2} = \frac{2}{x+10}\)[/tex] step by step.
1. Identify the common denominator:
To add and equate the fractions, we first identify the common denominator for the terms on the left-hand side ([tex]\(\frac{1}{x+1}\)[/tex] and [tex]\(\frac{1}{x+2}\)[/tex]) and the right-hand side ([tex]\(\frac{2}{x+10}\)[/tex]) of the equation.
The common denominator for the left-hand side is [tex]\((x + 1)(x + 2)\)[/tex].
The common denominator for the right-hand side is not necessary to solve directly but it can be rewritten for comprehension.
2. Rewrite each term with the common denominator:
Let's rewrite each fraction over the common denominator:
[tex]\[
\frac{1}{x+1} = \frac{x+2}{(x+1)(x+2)}
\][/tex]
[tex]\[
\frac{1}{x+2} = \frac{x+1}{(x+1)(x+2)}
\][/tex]
3. Combine the fractions on the left-hand side:
Adding the fractions together:
[tex]\[
\frac{x+2}{(x+1)(x+2)} + \frac{x+1}{(x+1)(x+2)}
= \frac{(x+2) + (x+1)}{(x+1)(x+2)}
= \frac{2x + 3}{(x+1)(x+2)}
\][/tex]
4. Equate the expression to the right-hand side:
Now, we will equate this to the right-hand side fraction:
[tex]\[
\frac{2x + 3}{(x+1)(x+2)} = \frac{2}{x+10}
\][/tex]
5. Cross-multiply to solve for [tex]\( x \)[/tex]:
By cross-multiplying, we get:
[tex]\[
(2x + 3)(x + 10) = 2(x + 1)(x + 2)
\][/tex]
6. Expand both sides of the equation:
[tex]\(\text{Left-hand side:}\)[/tex]
[tex]\[
(2x + 3)(x + 10) = 2x(x + 10) + 3(x + 10)
= 2x^2 + 20x + 3x + 30
= 2x^2 + 23x + 30
\][/tex]
[tex]\(\text{Right-hand side:}\)[/tex]
[tex]\[
2(x + 1)(x + 2) = 2(x^2 + 2x + 1x + 2)
= 2(x^2 + 3x + 2)
= 2x^2 + 6x + 4
\][/tex]
7. Set the expanded forms equal to each other:
Equate both simplified expressions:
[tex]\[
2x^2 + 23x + 30 = 2x^2 + 6x + 4
\][/tex]
8. Solve the resulting linear equation:
Subtract [tex]\(2x^2\)[/tex] from both sides:
[tex]\[
23x + 30 = 6x + 4
\][/tex]
Subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
17x + 30 = 4
\][/tex]
Subtract 30 from both sides:
[tex]\[
17x = -26
\][/tex]
Divide both sides by 17:
[tex]\[
x = -\frac{26}{17}
\][/tex]
So, the solution to the equation [tex]\(\frac{1}{x+1} + \frac{1}{x+2} = \frac{2}{x+10}\)[/tex] is [tex]\(x = -\frac{26}{17}\)[/tex].
