Respuesta :
Answer:
Any proposed solution of a rational equation that causes a denominator to equal __ZERO__ is rejected.
Step-by-step explanation:
We will show this statement is true by an example:
Consider the expression : [tex]\frac{x}{x-4}=\frac{x}{x-4}+4[/tex]
Now, solved the rational expression and check its proposed solution
[tex]\frac{x}{x-4}=\frac{x}{x-4}+4[/tex]
x cannot equal to 4, as it makes both denominators equal to zero.
Multiply both the sides by (x-4),
[tex]\left ( x-4 \right )\cdot \frac{x}{x-4}=\left (x-4 \right )\cdot\left ( \frac{x}{x-4}+4 \right )\\[/tex]
Now, use the distributive property on Right hand side,
[tex]\left ( x-4 \right )\cdot \frac{x}{x-4}=\left (x-4 \right )\cdot \left ( \frac{x}{x-4} \right )+\left (x-4 \right )\cdot 4\\[/tex]
Simplify the above expression,
[tex]x=x+4x-16[/tex]
Combine like terms,
[tex]x-x-4x=-16[/tex]
[tex]-4x=-16[/tex]
Divide both sides by -4, we get
[tex]\frac{-4x}{-4}=\frac{-16}{-4}[/tex]
[tex]x=4[/tex].
As we know that x cannot equal to 4, replacing x=4 in the original expression causes the denominator equal to 0.
Check the solution: [tex]\frac{x}{x-4}=\frac{x}{x-4}+4[/tex]
Substitute the value of x=4 in the original expression,
[tex]\frac{4}{4-4}=\frac{4}{4-4}+4[/tex]
[tex]\frac{4}{0}=\frac{4}{0}+4[/tex]
Thus, 4 must be rejected as the solution, and the solution set is only 0.
