The following rational equation has denominators that restrict the variable:

[tex]\frac{4}{x+3} + \frac{3}{x-4} = \frac{21}{(x+3)(x-4)}[/tex]

Identify the restrictions on the variable.



Answer :

To solve the given rational equation:

[tex]\[ \frac{4}{x+3}+\frac{3}{x-4}=\frac{21}{(x+3)(x-4)} \][/tex]

we need to determine the restrictions on the variable [tex]\( x \)[/tex].

### Step-by-Step Solution:

1. Identify the Denominators:
- The denominators are [tex]\( x + 3 \)[/tex] and [tex]\( x - 4 \)[/tex].

2. Find When Denominators Are Zero:
- The values of [tex]\( x \)[/tex] that make [tex]\( x + 3 \)[/tex] and [tex]\( x - 4 \)[/tex] equal to zero will cause the denominators to be undefined.

3. Set Each Denominator Equal to Zero and Solve:
- For [tex]\( x + 3 = 0 \)[/tex]:
[tex]\[ x + 3 = 0 \implies x = -3 \][/tex]
- For [tex]\( x - 4 = 0 \)[/tex]:
[tex]\[ x - 4 = 0 \implies x = 4 \][/tex]

### Conclusion:

The variable [tex]\( x \)[/tex] cannot take the values that make the denominators zero. Hence, the restrictions on the variable [tex]\( x \)[/tex] are:

[tex]\[ x \neq -3 \quad \text{and} \quad x \neq 4 \][/tex]

Therefore, the values of [tex]\( x \)[/tex] that cause the denominators to be zero and must be excluded from the solution are [tex]\(-3\)[/tex] and [tex]\(4\)[/tex].