Let f(x) = 1/(x+7)+15/(x-8). Find all a for which f(a)=f(a+15).
I understand this so far
f(a)=f(a+15)
1/(7+a) + 15/(a-8) = 1/(a+22) + 15/(a+7)
I have multiplied the LCD of (a-8)(a+7)(a+22)by both sides with no luck.
This is a multiple choice question
a.) -161/8
b.) 120/7
c.) 161/7
d.) -17
I have the answer as
Just plug in the values since it's multiple choice.
When I plug in the values what am I looking to find?
If you want to plug in, plug in the answer for a to see if both sides are equal.
you want 1/(7+a) + 15/(a-8) = 1/(a+22) + 15/(a+7), your second equation, to be true, so you true the four values of a given in the equation and if both sides are equal, you've found it.
*try not true
So you are saying that it is none of the multiple choice answers of -161/8
120/7 161/7 or -17
?
To solve the equation 1/(7+a) + 15/(a-8) = 1/(a+22) + 15/(a+7), we need to find the common denominator and then simplify the equation.
First, find the common denominator by multiplying all the denominators together: (a-8)(a+7)(a+22).
Multiply both sides of the equation by this common denominator to eliminate the fractions:
[(a-8)(a+7)(a+22)] * [1/(7+a) + 15/(a-8)] = [(a-8)(a+7)(a+22)] * [1/(a+22) + 15/(a+7)].
Simplifying this equation gives:
(a+7)(a+22) + 15(7+a) = (a-8)(a+7)(a+22) + 15(a+22).
Expand and simplify both sides of the equation:
(a^2 + 29a + 154) + (15a + 105) = (a^2 - a - 176)(a + 22) + 15(a + 22).
Combine like terms:
a^2 + 44a + 259 = a^3 + 21a^2 - 55a - 528 + 15a + 330.
Rearrange the equation and combine like terms to form a cubic equation:
a^3 + 21a^2 - 97a - 497 = 0.
To find the values of a that satisfy this cubic equation, you can use a variety of methods such as factoring, synthetic division, or the rational root theorem. Once you have found the roots of the cubic equation, you can check which of them satisfy the original equation f(a) = f(a+15).
However, you mentioned that this is a multiple-choice question and provided the answer options (a.) -161/8, b.) 120/7, c.) 161/7, d.) -17). To find the correct answer among the options, you can substitute each option into the original equation and check if f(a) = f(a+15). The option that satisfies the equation will be the correct answer.