$\frac{4}{x+16}-\frac{3}{x-16}=\frac{5 x}{x^2-256}$
a. Write the value or values of the variable that make a denominator zero.
$x=$ $\square$ (Use a comma to separate answers as needed.)
b. What is the solution of the equation? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is $\square$ 3. (Use a comma to separate answers as needed.).
B. The solution set is ${x \mid x$ is a real number}.
C. The solution set is $\varnothing$.
Answer (2)
Find the values of x that make the denominators zero: x = − 16 , 16 .
Multiply both sides of the equation by ( x + 16 ) ( x − 16 ) to eliminate the denominators.
Simplify the equation and solve for x , which gives x = − 28 .
Check for extraneous solutions. Since x = − 28 is not an extraneous solution, the solution set is { − 28 } .
Explanation
Find values that make denominators zero First, we need to identify the values of x that make the denominators zero. The denominators are x + 16 , x − 16 , and x 2 − 256 .
Solve for x We solve x + 16 = 0 to get x = − 16 . We solve x − 16 = 0 to get x = 16 . We solve x 2 − 256 = 0 to get x 2 = 256 , so x = ± 16 . Thus, the values that make the denominators zero are x = − 16 and x = 16 .
Rewrite the equation Now, we solve the equation x + 16 4 − x − 16 3 = x 2 − 256 5 x . We can rewrite the equation as x + 16 4 − x − 16 3 = ( x + 16 ) ( x − 16 ) 5 x .
Eliminate denominators Multiply both sides by ( x + 16 ) ( x − 16 ) to eliminate the denominators: 4 ( x − 16 ) − 3 ( x + 16 ) = 5 x .
Expand and simplify Expand and simplify: 4 x − 64 − 3 x − 48 = 5 x , which simplifies to x − 112 = 5 x .
Isolate x Subtract x from both sides: − 112 = 4 x .
Solve for x Divide by 4: x = − 28 .
Check for extraneous solutions We need to check if x = − 28 is an extraneous solution. Since − 28 = 16 and − 28 = − 16 , it is not an extraneous solution.
Final solution Therefore, the solution set is { − 28 } .
Examples
Rational equations appear in various fields, such as physics and engineering, when dealing with rates, proportions, and inverse relationships. For example, when calculating the combined resistance of parallel resistors in an electrical circuit, the formula involves rational expressions. Solving such equations helps determine the overall resistance, which is crucial for designing and analyzing circuits. Similarly, in fluid dynamics, rational equations can model flow rates and pressure relationships in pipes or channels, aiding in the design of efficient hydraulic systems.
The values of x that make the denominators zero are − 16 and 16 . The solution of the equation is { -28 }, so the correct choice is A. The solution set is { -28 }.
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