Any proposed solution of a rational equation that

Ответ:

Siris420

09.01.2022

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 : $\frac{x}{x-4}=\frac{x}{x-4}+4$

Now, solved the rational expression and check its proposed solution

$\frac{x}{x-4}=\frac{x}{x-4}+4$

x cannot equal to 4, as it makes both denominators equal to zero.

Multiply both the sides by (x-4),

$\left ( x-4 \right )\cdot \frac{x}{x-4}=\left (x-4 \right )\cdot\left ( \frac{x}{x-4}+4 \right )\\$

Now, use the distributive property on Right hand side,

$\left ( x-4 \right )\cdot \frac{x}{x-4}=\left (x-4 \right )\cdot \left ( \frac{x}{x-4} \right )+\left (x-4 \right )\cdot 4\\$

Simplify the above expression,

x=x+4x-16

Combine like terms,

x-x-4x=-16

-4x=-16

Divide both sides by -4, we get

$\frac{-4x}{-4}=\frac{-16}{-4}$

x=4 .

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: $\frac{x}{x-4}=\frac{x}{x-4}+4$

Substitute the value of x=4 in the original expression,

$\frac{4}{4-4}=\frac{4}{4-4}+4$

$\frac{4}{0}=\frac{4}{0}+4$

Thus, 4 must be rejected as the solution, and the solution set is only 0.

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