Problem 23 - Algebra 1

Question

Solve the equation by multiplying each side by the least common denominator. $$\frac{1}{x-4}+\frac{1}{x+4}=\frac{22}{x^{2}-16}$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 11 \)

Step 1: Identify the Least Common Denominator (LCD)

Look at the denominators (x-4), (x+4) and (x^2 - 16). Here, the denominator of the last fraction can be factored as (x-4)(x+4). Thus, the LCD of all the fractions will be (x-4)(x+4) or x^2 - 16.

Step 2: Multiply Each Term by the LCD

Multiply each term in the equation by the LCD, x^2 - 16: (x^2 - 16) * 1/(x-4) + (x^2 - 16) * 1/(x+4) = (x^2 - 16) * 22/(x^2 - 16).This eliminates the fractions and simplifies the equation to :(x + 4) + (x - 4) = 22.

Step 3: Simplify and Write in Standard Form

Simplify the equation to 2x = 22. Then, write it in standard form, which is 2x - 22 = 0.

Step 4: Solve for x

Solve the equation for x. Divide each side by 2 to get x = 11. However, we must make sure this solution doesn't make any denominator of the original fractions equal zero. Since 11 - 4 \neq 0 and 11 + 4 \neq 0, therefore x = 11 is a valid solution.