To solve the given equation, we need to address two parts: finding the restrictions on the variable and solving the equation while keeping these restrictions in mind.
Part a: Finding the restrictions on the variable
The given equation is:
[tex]\[
\frac{9}{7x + 21} = \frac{9}{x + 3} - \frac{6}{7}
\][/tex]
First, we need to determine the values of [tex]\(x\)[/tex] that cause the denominators to be zero.
1. For the denominator [tex]\(7x + 21\)[/tex]:
[tex]\[
7x + 21 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
7x = -21
\][/tex]
[tex]\[
x = -3
\][/tex]
2. For the denominator [tex]\(x + 3\)[/tex]:
[tex]\[
x + 3 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x = -3
\][/tex]
So, the value of the variable that makes the denominators zero is [tex]\(x = -3\)[/tex]. This is a restriction on the variable because at [tex]\(x = -3\)[/tex], at least one of the denominators becomes zero, which would make the expression undefined.
Restrictions:
[tex]\[
x = -3
\][/tex]
Part b: Solving the equation
Next, we solve the equation while keeping the restriction [tex]\(x \neq -3\)[/tex] in mind.
The equation to solve is:
[tex]\[
\frac{9}{7x + 21} = \frac{9}{x + 3} - \frac{6}{7}
\][/tex]
First, simplify the left-hand side and the right-hand side of the equation.
Left-hand side:
[tex]\[
\frac{9}{7x + 21}
\][/tex]
Factor the denominator:
[tex]\[
7x + 21 = 7(x + 3)
\][/tex]
So,
[tex]\[
\frac{9}{7(x + 3)}
\][/tex]
Right-hand side:
[tex]\[
\frac{9}{x + 3} - \frac{6}{7}
\][/tex]
To solve this equation, let's first eliminate the denominators by finding a common denominator. The common denominator is [tex]\(7(x + 3)\)[/tex].
Multiply every term by this common denominator:
[tex]\[
7(x + 3) \cdot \frac{9}{7(x + 3)} = 7(x + 3) \cdot \left(\frac{9}{x + 3} - \frac{6}{7}\right)
\][/tex]
This simplifies to:
[tex]\[
9 = 7 \cdot 9 - 6(x + 3)
\][/tex]
Simplify the right-hand side:
[tex]\[
9 = 63 - 6(x + 3)
\][/tex]
[tex]\[
9 = 63 - 6x - 18
\][/tex]
Combine like terms:
[tex]\[
9 = 45 - 6x
\][/tex]
To solve for [tex]\(x\)[/tex], isolate [tex]\(x\)[/tex]:
[tex]\[
9 - 45 = -6x
\][/tex]
[tex]\[
-36 = -6x
\][/tex]
[tex]\[
x = 6
\][/tex]
Valid solution:
We must ensure that our solution [tex]\(x = 6\)[/tex] does not violate the restriction [tex]\(x \neq -3\)[/tex]. Since [tex]\(x = 6\)[/tex] is not in the restricted values, it is a valid solution.
Answer:
[tex]\[
x = 6
\][/tex]
