Question

7 fill in the blank 1 point
what is the solution of the following equation? $\frac{14}{x+2}-6=\frac{2}{x+2}$
enter your answer in the space provided.

Explanation:

Step1: Eliminate denominators

Multiply all terms by $x+2$ (where $x
eq -2$):
$$14 - 6(x+2) = 2$$

Step2: Expand the left side

$$14 - 6x - 12 = 2$$

Step3: Simplify constants

Combine like terms:
$$2 - 6x = 2$$

Step4: Isolate the variable term

Subtract 2 from both sides:
$$-6x = 0$$

Step5: Solve for x

Divide by -6:
$$x = 0 \div (-6) = -1$$
(Verify: Substitute $x=-1$ into original equation: $\frac{14}{-1+2}-6=14-6=8$, $\frac{2}{-1+2}=2$? Correction: Step 3 error correction:

Step1: Eliminate denominators

Multiply all terms by $x+2$ ($x
eq -2$):
$$14 - 6(x+2) = 2$$

Step2: Expand left side

$$14 - 6x - 12 = 2$$

Step3: Simplify left side

$$2 - 6x = 2$$

Step4: Subtract 2 from both sides

$$-6x = 0$$

Step5: Correct simplification fix

Wait, error in calculation: $14-12=2$, so $2-6x=2$ → $-6x=0$ is wrong. Correct Step 3:
$$14 - 6x -12 = 2 \implies 2 - 6x = 2 \implies -6x = 2-2=0$$
No, wait original equation: $\frac{14}{x+2}-6=\frac{2}{x+2}$
Move $\frac{2}{x+2}$ to left: $\frac{14-2}{x+2}-6=0 \implies \frac{12}{x+2}=6$

Step1: Rewrite equation by moving terms

$$\frac{14-2}{x+2}=6$$

Step2: Simplify numerator

$$\frac{12}{x+2}=6$$

Step3: Solve for denominator

$$x+2=\frac{12}{6}=2$$

Step4: Find x

$$x=2-2=0$$
(Verification: $\frac{14}{0+2}-6=7-6=1$, $\frac{2}{0+2}=1$. Correct solution is $x=0$)

Corrected Answer:

$x=0$

Step1: Combine fraction terms

Move $\frac{2}{x+2}$ to left side:
$$\frac{14}{x+2}-\frac{2}{x+2}=6$$

Step2: Simplify left fraction

$$\frac{14-2}{x+2}=6 \implies \frac{12}{x+2}=6$$

Step3: Isolate denominator

Multiply both sides by $x+2$, divide by 6:
$$x+2=\frac{12}{6}=2$$

Step4: Solve for x

Subtract 2 from both sides:
$$x=2-2=0$$

Answer:

$x=-1$