Solve for $$ x $$. $$ -\frac{7}{x+3}=\frac{1}{4 x+12}-3 $$ If there is more than one solution, separate them with commas. If there is no solution, dick on "No solution". $$ x=[] $$

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UpStudy AI · Accepted Answer

1. Rewrite the equation:
   $$
   -\frac{7}{x+3}=\frac{1}{4x+12}-3
   $$
   Note that:
   $$
   4x+12=4(x+3)
   $$
   So the equation becomes:
   $$
   -\frac{7}{x+3}=\frac{1}{4(x+3)}-3
   $$

2. Let $$ t = x+3 $$. Substitute into the equation:
   $$
   -\frac{7}{t}=\frac{1}{4t}-3
   $$
   (Also note $$ t \neq 0 $$ since $$ x \neq -3 $$).

3. Multiply the whole equation by $$ 4t $$ (which is allowed because $$ t \neq 0 $$):
   $$
   4t \left(-\frac{7}{t}\right) = 4t \left(\frac{1}{4t}\right) - 3\cdot 4t
   $$
   Simplify each term:
   $$
   -28 = 1 - 12t
   $$

4. Solve for $$ t $$:
   $$
   -28 = 1 - 12t
   $$
   Subtract $$ 1 $$ from both sides:
   $$
   -29 = -12t
   $$
   Divide both sides by $$-12$$:
   $$
   t = \frac{29}{12}
   $$

5. Substitute back $$ t = x+3 $$:
   $$
   x+3=\frac{29}{12}
   $$
   Solve for $$ x $$ by subtracting $$3$$:
   $$
   x = \frac{29}{12} - 3
   $$
   Convert $$ 3 $$ to a fraction with denominator $$ 12 $$:
   $$
   x = \frac{29}{12} - \frac{36}{12} = -\frac{7}{12}
   $$

6. The solution is:
   $$
   x = -\frac{7}{12}
   $$
$$ x = -\frac{7}{12} $$

Solve for .

If there is more than one solution, separate them with commas.
If there is no solution, dick on “No solution”.

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Solve for . If there is more than one solution, separate them with commas. If there is no solution, dick on “No solution”.