Use the theorem for absolute value equations along with the techniques in Section 0.8 to find all real solutions. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.)

Absvalue(7*x / x*2 − 1) = 7

To solve the equation:

\[
| \frac{7x}{x^2 - 1} | = 7
\]

we use the property of absolute value equations:

\[
|A| = B \implies A = B \text{ or } A = -B
\]

### Step 1: Remove the absolute value
We write two cases:
1. \( \frac{7x}{x^2 - 1} = 7 \)
2. \( \frac{7x}{x^2 - 1} = -7 \)

#### Case 1: \( \frac{7x}{x^2 - 1} = 7 \)
Multiply both sides by \( x^2 - 1 \) to eliminate the denominator:

\[
7x = 7(x^2 - 1)
\]

Expand the right-hand side:

\[
7x = 7x^2 - 7
\]

Move all terms to one side:

\[
0 = 7x^2 - 7x - 7
\]

Divide by 7:

\[
x^2 - x - 1 = 0
\]

Solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}
\]

\[
x = \frac{1 \pm \sqrt{1 + 4}}{2}
\]

\[
x = \frac{1 \pm \sqrt{5}}{2}
\]

So, the solutions for Case 1 are:

\[
x = \frac{1 + \sqrt{5}}{2}, \quad x = \frac{1 - \sqrt{5}}{2}
\]

#### Case 2: \( \frac{7x}{x^2 - 1} = -7 \)
Multiply both sides by \( x^2 - 1 \):

\[
7x = -7(x^2 - 1)
\]

Expand the right-hand side:

\[
7x = -7x^2 + 7
\]

Move all terms to one side:

\[
0 = 7x^2 + 7x - 7
\]

Divide by 7:

\[
x^2 + x - 1 = 0
\]

Solve this quadratic equation using the quadratic formula:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)}
\]

\[
x = \frac{-1 \pm \sqrt{1 + 4}}{2}
\]

\[
x = \frac{-1 \pm \sqrt{5}}{2}
\]

So, the solutions for Case 2 are:

\[
x = \frac{-1 + \sqrt{5}}{2}, \quad x = \frac{-1 - \sqrt{5}}{2}
\]

### Step 2: Check for extraneous solutions
The expression \( \frac{7x}{x^2 - 1} \) has restrictions: \( x^2 - 1 \

Use the theorem for absolute value equations along with the techniques in Section 0.8 to find all real solutions. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) Absvalue(7*x / x^2 − 1) = 7

Learn how to solve the absolute value equation |7x / (x^2 - 1)| = 7. This step-by-step guide covers key mathematical concepts such as absolute value equations, rational expressions, and the quadratic formula. Suitable for high school students, Grades 9-12.

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