Respuesta :
Answer:
A. -4 and 4
B. No solution.
Step-by-step explanation:
The given equation is
[tex]\dfrac{3}{x+4}+\dfrac{2}{x-4}=\dfrac{16}{(x+4)(x-4)}[/tex]
A.
Equate the denominators equal to 0 to find the restrictions on the variable.
[tex]x+4=0\Rightarrow x=-4[/tex]
[tex]x-4=0\Rightarrow x=4[/tex]
Therefore, [tex]x\neq -4,4[/tex].
B.
We have,
[tex]\dfrac{3}{x+4}+\dfrac{2}{x-4}=\dfrac{16}{(x+4)(x-4)}[/tex]
[tex]\dfrac{3(x-4)+2(x+4)}{(x+4)(x-4)}=\dfrac{16}{(x+4)(x-4)}[/tex]
Multiply both sides by (x-4)(x+4).
[tex]3x-12+2x+8=16[/tex]
[tex]5x-4=16[/tex]
Add 4 on both sides.
[tex]5x=16+4[/tex]
[tex]5x=20[/tex]
Divide both sides by 5.
[tex]x=4[/tex]
Here the solution is x=4 but it is the restricted value.
Therefore, the given equation has no solution.
For the given rational equation, we have that:
- A. The values of the variable that make(s) the denominators zero are x = -4 and x = 4.
- B. The equation has no solution.
Rational Equation:
The rational equation given in this problem is:
[tex]\frac{3}{x + 4} + \frac{2}{x - 4} = \frac{16}{(x + 4)(x - 4)}[/tex]
Then, applying the least common factor:
[tex]\frac{3(x - 4) + 2(x + 4)}{(x + 4)(x - 4)} = \frac{16}{(x + 4)(x - 4)}[/tex]
[tex]\frac{5x - 4}{(x + 4)(x - 4)} - \frac{16}{(x + 4)(x - 4)} = 0[/tex]
[tex]\frac{5x - 20}{(x + 4)(x - 4)} = 0[/tex]
Item a:
The denominator cannot be zero, hence:
[tex](x + 4)(x - 4) \neq 0[/tex]
[tex]x \neq -4[/tex]
[tex]x \neq 4[/tex]
The values of the variable that make(s) the denominators zero are x = -4 and x = 4.
Item b:
[tex]5x - 20 = 0[/tex]
[tex]5x = 20[/tex]
[tex]x = \frac{20}{5}[/tex]
[tex]x = 4[/tex]
However, x = 4 makes the denominator zero, hence the equation has no solution.
You can learn more about rational functions at https://brainly.com/question/13136492
